diff --git a/mypy/plugins/dataclasses.py b/mypy/plugins/dataclasses.py
--- a/mypy/plugins/dataclasses.py
+++ b/mypy/plugins/dataclasses.py
@@ -4,9 +4,10 @@
 from typing_extensions import Final
 
 from mypy.nodes import (
-    ARG_OPT, ARG_NAMED, ARG_NAMED_OPT, ARG_POS, MDEF, Argument, AssignmentStmt, CallExpr,
-    Context, Expression, JsonDict, NameExpr, RefExpr,
-    SymbolTableNode, TempNode, TypeInfo, Var, TypeVarExpr, PlaceholderNode
+    ARG_OPT, ARG_NAMED, ARG_NAMED_OPT, ARG_POS, ARG_STAR, ARG_STAR2, MDEF,
+    Argument, AssignmentStmt, CallExpr,    Context, Expression, JsonDict,
+    NameExpr, RefExpr, SymbolTableNode, TempNode, TypeInfo, Var, TypeVarExpr,
+    PlaceholderNode
 )
 from mypy.plugin import ClassDefContext, SemanticAnalyzerPluginInterface
 from mypy.plugins.common import (
@@ -141,11 +142,28 @@ def transform(self) -> None:
         if (decorator_arguments['init'] and
                 ('__init__' not in info.names or info.names['__init__'].plugin_generated) and
                 attributes):
+
+            args = [attr.to_argument() for attr in attributes if attr.is_in_init
+                    and not self._is_kw_only_type(attr.type)]
+
+            if info.fallback_to_any:
+                # Make positional args optional since we don't know their order.
+                # This will at least allow us to typecheck them if they are called
+                # as kwargs
+                for arg in args:
+                    if arg.kind == ARG_POS:
+                        arg.kind = ARG_OPT
+
+                nameless_var = Var('')
+                args = [Argument(nameless_var, AnyType(TypeOfAny.explicit), None, ARG_STAR),
+                        *args,
+                        Argument(nameless_var, AnyType(TypeOfAny.explicit), None, ARG_STAR2),
+                        ]
+
             add_method(
                 ctx,
                 '__init__',
-                args=[attr.to_argument() for attr in attributes if attr.is_in_init
-                      and not self._is_kw_only_type(attr.type)],
+                args=args,
                 return_type=NoneType(),
             )
 

diff --git a/test-data/unit/check-dataclasses.test b/test-data/unit/check-dataclasses.test
--- a/test-data/unit/check-dataclasses.test
+++ b/test-data/unit/check-dataclasses.test
@@ -1522,3 +1522,17 @@ PublishedMessagesVar = dict[int, 'PublishedMessages']
 class PublishedMessages:
     left: int
 [builtins fixtures/dataclasses.pyi]
+
+[case testDataclassesAnyInherit]
+# flags: --python-version 3.7
+from dataclasses import dataclass
+from typing import Any
+B: Any
+@dataclass
+class A(B):
+    a: int
+
+A(a=1, b=2)
+A(1)
+A(a="foo")  # E: Argument "a" to "A" has incompatible type "str"; expected "int"
+[builtins fixtures/dataclasses.pyi]

Inheriting from Any with dataclass
I'm importing a package at a path mypy doesn't know about and using the `--ignore-missing-imports` flag.

```python a.py
import dataclasses
from mypackage import MyClass

# @dataclasses.dataclass
# class MyClass:
#     bar: str

@dataclasses.dataclass
class Inherits(MyClass):
    foo: str

reveal_type(MyClass)
reveal_type(Inherits)

i = Inherits(foo="hi", bar="bye")
```

The type of MyClass is Any because mypy doesn't know what it is. I would expect a class that inherits from Any to also be Any or at least be able to take any number of arguments in its constructor. Instead I get:

```
a.py:12: note: Revealed type is "Any"
a.py:13: note: Revealed type is "def (foo: builtins.str) -> a.Inherits"
a.py:15: error: Unexpected keyword argument "bar" for "Inherits"
Found 1 error in 1 file (checked 1 source file)
```

mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesCustomInit mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesBasic mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesBasicInheritance mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesDeepInheritance mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesOverridingWithDefaults mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesOverriding mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesDeepInitVarInheritance mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesMultipleInheritance mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesFreezing mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesMultiInit mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesFields mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesBadInit mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesDefaultFactoryTypeChecking mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesDefaults mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesDefaultFactories mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesMultiInitDefaults mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesOrderingKwOnlyOnFieldFalse mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesOrderingKwOnlyOnField mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesOrderingKwOnly mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesDefaultOrdering mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesClassmethods mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassOrderingWithoutEquality mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesOrderingKwOnlyWithSentinel mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassUntypedGenericInheritance mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassOrderingWithCustomMethods mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesOrderingKwOnlyWithSentinelAndFieldOverride mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesOverloadsAndClassmethods mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassGenericSubtype mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesClassVars mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassDefaultsInheritance mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesOrderingKwOnlyWithSentinelAndSubclass mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassGenericInheritance mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesOrderingKwOnlyWithMultipleSentinel mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassMultiGenericInheritance mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassOrdering mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassGenerics mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesNoInitInitVarInheritance mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testNoComplainFieldNone mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesInitVarNoOverride mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassGenericsClassmethod mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesForwardRefs mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassFactory mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testNoComplainFieldNoneStrict mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesInitVarPostInitOverride mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesInitVars mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDisallowUntypedWorksForward mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesInitVarOverride mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDisallowUntypedWorksForwardBad mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesInitVarsAndDefer mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassOrderingDeferred mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesInitVarIncremental mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassFieldDeferredFrozen mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDeferredDataclassInitSignature mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassFieldDeferred mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testMemberExprWorksAsField mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesDefaultsMroOtherFile mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDeferredDataclassInitSignatureSubclass mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testTypeInDataclassDeferredStar mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesDefaultsIncremental mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassInheritanceNoAnnotation2 mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassInheritanceNoAnnotation mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesInheritingDuplicateField mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassHasAttributeWithFields mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassCallableAssignment mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassCallableProperty mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassCallablePropertyWrongTypeAssignment mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassCallablePropertyWrongType mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassFieldWithTypedDictUnpacking mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassFieldDoesNotFailOnKwargsUnpacking mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassFieldWithPositionalArguments mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassWithSlotsArg mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testSlotsDefinitionWithTwoPasses1 mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassWithSlotsConflict mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassWithSlotsDef mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassWithSlotsArgBefore310 mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testSlotsDefinitionWithTwoPasses2 mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testSlotsDefinitionWithTwoPasses4 mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testSlotsDefinitionWithTwoPasses3
mypy/test/testcheck.py::TypeCheckSuite::check-dataclasses.test::testDataclassesAnyInherit