diff --git a/CHANGES.rst b/CHANGES.rst
index eb46957df..efb20fed8 100644
--- a/CHANGES.rst
+++ b/CHANGES.rst
@@ -8,6 +8,7 @@ Unreleased
 -   Use modern packaging metadata with ``pyproject.toml`` instead of ``setup.cfg``.
     :pr:`1793`
 -   Use ``flit_core`` instead of ``setuptools`` as build backend.
+-   Avoid unclosed ``auto_aiter`` warnings. :pr:`1954`
 
 
 Version 3.1.3
diff --git a/src/jinja2/async_utils.py b/src/jinja2/async_utils.py
index e65219e49..b0d277de7 100644
--- a/src/jinja2/async_utils.py
+++ b/src/jinja2/async_utils.py
@@ -6,6 +6,9 @@
 from .utils import _PassArg
 from .utils import pass_eval_context
 
+if t.TYPE_CHECKING:
+    import typing_extensions as te
+
 V = t.TypeVar("V")
 
 
@@ -67,15 +70,27 @@ async def auto_await(value: t.Union[t.Awaitable["V"], "V"]) -> "V":
     return t.cast("V", value)
 
 
-async def auto_aiter(
+class _IteratorToAsyncIterator(t.Generic[V]):
+    def __init__(self, iterator: "t.Iterator[V]"):
+        self._iterator = iterator
+
+    def __aiter__(self) -> "te.Self":
+        return self
+
+    async def __anext__(self) -> V:
+        try:
+            return next(self._iterator)
+        except StopIteration as e:
+            raise StopAsyncIteration(e.value) from e
+
+
+def auto_aiter(
     iterable: "t.Union[t.AsyncIterable[V], t.Iterable[V]]",
 ) -> "t.AsyncIterator[V]":
     if hasattr(iterable, "__aiter__"):
-        async for item in t.cast("t.AsyncIterable[V]", iterable):
-            yield item
+        return iterable.__aiter__()
     else:
-        for item in iterable:
-            yield item
+        return _IteratorToAsyncIterator(iter(iterable))
 
 
 async def auto_to_list(
diff --git a/src/jinja2/compiler.py b/src/jinja2/compiler.py
index ff95c807b..5551b9455 100644
--- a/src/jinja2/compiler.py
+++ b/src/jinja2/compiler.py
@@ -973,14 +973,18 @@ def visit_Block(self, node: nodes.Block, frame: Frame) -> None:
                 f"yield from context.blocks[{node.name!r}][0]({context})", node
             )
         else:
+            self.writeline("gen = context.blocks[{node.name!r}][0]({context})")
+            self.writeline("try:")
+            self.indent()
             self.writeline(
-                f"{self.choose_async()}for event in"
-                f" context.blocks[{node.name!r}][0]({context}):",
+                f"{self.choose_async()}for event in gen:"
                 node,
             )
             self.indent()
             self.simple_write("event", frame)
             self.outdent()
+            self.outdent()
+            self.write_line(f"finally: {self.choose_async('await gen.aclose()', 'gen.close()')}")
 
         self.outdent(level)