diff --git a/src/quart/utils.py b/src/quart/utils.py
index 7e48ace..a92fc03 100644
--- a/src/quart/utils.py
+++ b/src/quart/utils.py
@@ -10,13 +10,14 @@
 from pathlib import Path
 from typing import (
     Any,
-    AsyncGenerator,
+    AsyncIterator,
     Awaitable,
     Callable,
     Coroutine,
-    Generator,
     Iterable,
+    Iterator,
     TYPE_CHECKING,
+    TypeVar,
 )
 
 from werkzeug.datastructures import Headers
@@ -66,12 +67,15 @@ async def _wrapper(*args: Any, **kwargs: Any) -> Any:
     return _wrapper
 
 
-def run_sync_iterable(iterable: Generator[Any, None, None]) -> AsyncGenerator[Any, None]:
-    async def _gen_wrapper() -> AsyncGenerator[Any, None]:
+T = TypeVar("T")
+
+
+def run_sync_iterable(iterable: Iterator[T]) -> AsyncIterator[T]:
+    async def _gen_wrapper() -> AsyncIterator[T]:
         # Wrap the generator such that each iteration runs
         # in the executor. Then rationalise the raised
         # errors so that it ends.
-        def _inner() -> Any:
+        def _inner() -> T:
             # https://bugs.python.org/issue26221
             # StopIteration errors are swallowed by the
             # run_in_exector method
diff --git a/src/quart/wrappers/response.py b/src/quart/wrappers/response.py
index ad2e6c5..01bf4c0 100644
--- a/src/quart/wrappers/response.py
+++ b/src/quart/wrappers/response.py
@@ -2,7 +2,6 @@
 
 from abc import ABC, abstractmethod
 from hashlib import md5
-from inspect import isasyncgen, isgenerator
 from io import BytesIO
 from os import PathLike
 from types import TracebackType
@@ -101,27 +100,21 @@ async def __anext__(self) -> bytes:
 
 
 class IterableBody(ResponseBody):
-    def __init__(self, iterable: AsyncGenerator[bytes, None] | Iterable) -> None:
-        self.iter: AsyncGenerator[bytes, None]
-        if isasyncgen(iterable):
-            self.iter = iterable
-        elif isgenerator(iterable):
-            self.iter = run_sync_iterable(iterable)
+    def __init__(self, iterable: AsyncIterable[Any] | Iterable[Any]) -> None:
+        self.iter: AsyncIterator[Any]
+        if isinstance(iterable, Iterable):
+            self.iter = run_sync_iterable(iter(iterable))
         else:
-
-            async def _aiter() -> AsyncGenerator[bytes, None]:
-                for data in iterable:  # type: ignore
-                    yield data
-
-            self.iter = _aiter()
+            self.iter = iterable.__aiter__()  # Can't use aiter() until 3.10
 
     async def __aenter__(self) -> IterableBody:
         return self
 
     async def __aexit__(self, exc_type: type, exc_value: BaseException, tb: TracebackType) -> None:
-        await self.iter.aclose()
+        if hasattr(self.iter, "aclose"):
+            await self.iter.aclose()
 
-    def __aiter__(self) -> AsyncIterator:
+    def __aiter__(self) -> AsyncIterator[Any]:
         return self.iter
 
 
@@ -261,7 +254,7 @@ class Response(SansIOResponse):
 
     def __init__(
         self,
-        response: ResponseBody | str | bytes | Iterable | None = None,
+        response: ResponseBody | str | bytes | Iterable | AsyncIterable | None = None,
         status: int | None = None,
         headers: dict | Headers | None = None,
         mimetype: str | None = None,
diff --git a/tests/test_templating.py b/tests/test_templating.py
index f745020..5df5403 100644
--- a/tests/test_templating.py
+++ b/tests/test_templating.py
@@ -9,6 +9,7 @@
     g,
     Quart,
     render_template_string,
+    Response,
     ResponseReturnValue,
     session,
     stream_template_string,
@@ -148,3 +149,11 @@ async def index() -> ResponseReturnValue:
     test_client = app.test_client()
     response = await test_client.get("/")
     assert (await response.data) == b"42"
+
+    @app.get("/2")
+    async def index2() -> ResponseReturnValue:
+        return Response(await stream_template_string("{{ config }}", config=43))
+
+    test_client = app.test_client()
+    response = await test_client.get("/2")
+    assert (await response.data) == b"43"