diff --git a/starlette/middleware/base.py b/starlette/middleware/base.py
index 49a5e3e2d..721a2eb58 100644
--- a/starlette/middleware/base.py
+++ b/starlette/middleware/base.py
@@ -4,7 +4,7 @@
 
 from starlette.requests import Request
 from starlette.responses import Response, StreamingResponse
-from starlette.types import ASGIApp, Receive, Scope, Send
+from starlette.types import ASGIApp, Message, Receive, Scope, Send
 
 RequestResponseEndpoint = typing.Callable[[Request], typing.Awaitable[Response]]
 DispatchFunction = typing.Callable[
@@ -28,12 +28,26 @@ async def call_next(request: Request) -> Response:
             app_exc: typing.Optional[Exception] = None
             send_stream, recv_stream = anyio.create_memory_object_stream()
 
+            async def send(msg: Message) -> None:
+                # Shield send "http.response.start" from cancellation.
+                # Otherwise, `await recv_stream.receive()` will raise
+                # `anyio.EndOfStream` if the connection is disconnected,
+                # due to `task_group.cancel_scope.cancel()` in
+                # `StreamingResponse.__call__.<locals>.wrap`
+                # and cancellation check during `await checkpoint()` in
+                # `MemoryObjectSendStream.send`.
+                # This would trigger the check we have in this middleware resulting in
+                # `RuntimeError: No response returned.` being raised below.
+                shield = msg["type"] == "http.response.start"
+                with anyio.CancelScope(shield=shield):
+                    await send_stream.send(msg)
+
             async def coro() -> None:
                 nonlocal app_exc
 
                 async with send_stream:
                     try:
-                        await self.app(scope, request.receive, send_stream.send)
+                        await self.app(scope, request.receive, send)
                     except Exception as exc:
                         app_exc = exc
 
diff --git a/tests/middleware/test_base.py b/tests/middleware/test_base.py
index 976d77b86..80087686d 100644
--- a/tests/middleware/test_base.py
+++ b/tests/middleware/test_base.py
@@ -1,13 +1,16 @@
 import contextvars
+from contextlib import AsyncExitStack
+from typing import AsyncGenerator, Awaitable, Callable
 
 import pytest
 
 from starlette.applications import Starlette
 from starlette.middleware import Middleware
 from starlette.middleware.base import BaseHTTPMiddleware
-from starlette.responses import PlainTextResponse, StreamingResponse
+from starlette.requests import Request
+from starlette.responses import PlainTextResponse, Response, StreamingResponse
 from starlette.routing import Route, WebSocketRoute
-from starlette.types import ASGIApp, Receive, Scope, Send
+from starlette.types import ASGIApp, Message, Receive, Scope, Send
 
 
 class CustomMiddleware(BaseHTTPMiddleware):
@@ -206,3 +209,41 @@ async def homepage(request):
     client = test_client_factory(app)
     response = client.get("/")
     assert response.status_code == 200, response.content
+
+
+@pytest.mark.anyio
+async def test_client_disconnects_before_response_is_sent() -> None:
+    app: ASGIApp
+
+    async def homepage(request: Request):
+        # await anyio.sleep(5)
+        return PlainTextResponse("hi!")
+
+    async def dispatch(
+        request: Request, call_next: Callable[[Request], Awaitable[Response]]
+    ) -> Response:
+        return await call_next(request)
+
+    app = BaseHTTPMiddleware(Route("/", homepage), dispatch=dispatch)
+    app = BaseHTTPMiddleware(app, dispatch=dispatch)
+
+    async def recv_gen() -> AsyncGenerator[Message, None]:
+        yield {"type": "http.request"}
+        yield {"type": "http.disconnect"}
+        yield {"type": "http.disconnect"}
+
+    async def send_gen() -> AsyncGenerator[None, Message]:
+        msg = yield
+        assert msg["type"] == "http.response.start"
+        msg = yield
+        raise AssertionError("Should not be called")  # pragma: no cover
+
+    scope = {"type": "http", "method": "GET", "path": "/"}
+
+    async with AsyncExitStack() as stack:
+        recv = recv_gen()
+        stack.push_async_callback(recv.aclose)
+        send = send_gen()
+        stack.push_async_callback(send.aclose)
+        await send.__anext__()
+        await app(scope, recv.__aiter__().__anext__, send.asend)