diff --git a/starlette/middleware/base.py b/starlette/middleware/base.py
index 77ba66925..423f40777 100644
--- a/starlette/middleware/base.py
+++ b/starlette/middleware/base.py
@@ -23,17 +23,25 @@ async def __call__(self, scope: Scope, receive: Receive, send: Send) -> None:
             return
 
         async def call_next(request: Request) -> Response:
+            app_exc: typing.Optional[Exception] = None
             send_stream, recv_stream = anyio.create_memory_object_stream()
 
             async def coro() -> None:
+                nonlocal app_exc
+
                 async with send_stream:
-                    await self.app(scope, request.receive, send_stream.send)
+                    try:
+                        await self.app(scope, request.receive, send_stream.send)
+                    except Exception as exc:
+                        app_exc = exc
 
             task_group.start_soon(coro)
 
             try:
                 message = await recv_stream.receive()
             except anyio.EndOfStream:
+                if app_exc is not None:
+                    raise app_exc
                 raise RuntimeError("No response returned.")
 
             assert message["type"] == "http.response.start"
diff --git a/tests/middleware/test_base.py b/tests/middleware/test_base.py
index 8a8df4ea6..c6bfd49fc 100644
--- a/tests/middleware/test_base.py
+++ b/tests/middleware/test_base.py
@@ -25,7 +25,7 @@ def homepage(request):
 
 @app.route("/exc")
 def exc(request):
-    raise Exception()
+    raise Exception("Exc")
 
 
 @app.route("/no-response")
@@ -52,8 +52,9 @@ def test_custom_middleware(test_client_factory):
     response = client.get("/")
     assert response.headers["Custom-Header"] == "Example"
 
-    with pytest.raises(Exception):
+    with pytest.raises(Exception) as ctx:
         response = client.get("/exc")
+    assert str(ctx.value) == "Exc"
 
     with pytest.raises(RuntimeError):
         response = client.get("/no-response")