diff --git a/starlette/middleware/base.py b/starlette/middleware/base.py
index 49a5e3e2d..2ea98a4bc 100644
--- a/starlette/middleware/base.py
+++ b/starlette/middleware/base.py
@@ -33,6 +33,7 @@ async def coro() -> None:
 
                 async with send_stream:
                     try:
+                        # `send_stream.send` blocks until `body_stream` is called
                         await self.app(scope, request.receive, send_stream.send)
                     except Exception as exc:
                         app_exc = exc
diff --git a/starlette/responses.py b/starlette/responses.py
index 98c8caf1b..c15fd2848 100644
--- a/starlette/responses.py
+++ b/starlette/responses.py
@@ -166,8 +166,9 @@ async def __call__(self, scope: Scope, receive: Receive, send: Send) -> None:
         )
         await send({"type": "http.response.body", "body": self.body})
 
-        if self.background is not None:
-            await self.background()
+        with anyio.CancelScope(shield=True):
+            if self.background is not None:
+                await self.background()
 
 
 class HTMLResponse(Response):
@@ -264,8 +265,9 @@ async def wrap(func: typing.Callable[[], typing.Coroutine]) -> None:
             task_group.start_soon(wrap, partial(self.stream_response, send))
             await wrap(partial(self.listen_for_disconnect, receive))
 
-        if self.background is not None:
-            await self.background()
+        with anyio.CancelScope(shield=True):
+            if self.background is not None:
+                await self.background()
 
 
 class FileResponse(Response):
@@ -350,5 +352,7 @@ async def __call__(self, scope: Scope, receive: Receive, send: Send) -> None:
                             "more_body": more_body,
                         }
                     )
-        if self.background is not None:
-            await self.background()
+
+        with anyio.CancelScope(shield=True):
+            if self.background is not None:
+                await self.background()
diff --git a/tests/middleware/test_base.py b/tests/middleware/test_base.py
index 976d77b86..c7b65e8e4 100644
--- a/tests/middleware/test_base.py
+++ b/tests/middleware/test_base.py
@@ -1,11 +1,13 @@
 import contextvars
 
+import anyio
 import pytest
 
 from starlette.applications import Starlette
+from starlette.background import BackgroundTasks
 from starlette.middleware import Middleware
 from starlette.middleware.base import BaseHTTPMiddleware
-from starlette.responses import PlainTextResponse, StreamingResponse
+from starlette.responses import PlainTextResponse, Response, StreamingResponse
 from starlette.routing import Route, WebSocketRoute
 from starlette.types import ASGIApp, Receive, Scope, Send
 
@@ -84,6 +86,34 @@ def test_custom_middleware(test_client_factory):
         assert text == "Hello, world!"
 
 
+def test_background_tasks(test_client_factory):
+    async def _sleep(identifier, delay):
+        print(identifier, "started")
+        await anyio.sleep(delay)
+        print(identifier, "completed")
+
+    def _sleep_sync(identifier, delay):
+        import time
+
+        print(identifier, "started")
+        time.sleep(delay)
+        print(identifier, "completed")
+
+    async def bg_task(request):
+        background_tasks = BackgroundTasks()
+        background_tasks.add_task(_sleep, "background task 1", 2)
+        background_tasks.add_task(_sleep, "background task 2", 2)
+        background_tasks.add_task(_sleep_sync, "background task sync", 2)
+        return Response(background=background_tasks)
+
+    app = Starlette(
+        routes=[Route("/bg-task", bg_task)], middleware=[Middleware(CustomMiddleware)]
+    )
+    client = test_client_factory(app)
+    response = client.get("/bg-task")
+    assert response.text == ""
+
+
 def test_state_data_across_multiple_middlewares(test_client_factory):
     expected_value1 = "foo"
     expected_value2 = "bar"
@@ -141,12 +171,25 @@ def homepage(request):
 
 def test_fully_evaluated_response(test_client_factory):
     # Test for https://github.com/encode/starlette/issues/1022
-    class CustomMiddleware(BaseHTTPMiddleware):
+    class ConsumeMiddleware(BaseHTTPMiddleware):
+        async def dispatch(self, request, call_next):
+            resp = await call_next(request)
+
+            async def _send(m):
+                pass
+
+            await resp.stream_response(_send)  # type: ignore
+
+            return PlainTextResponse("Custom")
+
+    class DiscardMiddleware(BaseHTTPMiddleware):
         async def dispatch(self, request, call_next):
             await call_next(request)
             return PlainTextResponse("Custom")
 
-    app = Starlette(middleware=[Middleware(CustomMiddleware)])
+    app = Starlette(
+        middleware=[Middleware(ConsumeMiddleware), Middleware(DiscardMiddleware)]
+    )
 
     client = test_client_factory(app)
     response = client.get("/does_not_exist")