diff --git a/httpx/_client.py b/httpx/_client.py
index c57cfb6ea9..3ba5370f68 100644
--- a/httpx/_client.py
+++ b/httpx/_client.py
@@ -901,7 +901,7 @@ def send(
 
             return response
 
-        except Exception as exc:
+        except BaseException as exc:
             response.close()
             raise exc
 
@@ -933,7 +933,7 @@ def _send_handling_auth(
                     request = next_request
                     history.append(response)
 
-                except Exception as exc:
+                except BaseException as exc:
                     response.close()
                     raise exc
         finally:
@@ -972,7 +972,7 @@ def _send_handling_redirects(
                     response.next_request = request
                     return response
 
-            except Exception as exc:
+            except BaseException as exc:
                 response.close()
                 raise exc
 
@@ -1605,7 +1605,7 @@ async def send(
 
             return response
 
-        except Exception as exc:  # pragma: no cover
+        except BaseException as exc:  # pragma: no cover
             await response.aclose()
             raise exc
 
@@ -1637,7 +1637,7 @@ async def _send_handling_auth(
                     request = next_request
                     history.append(response)
 
-                except Exception as exc:
+                except BaseException as exc:
                     await response.aclose()
                     raise exc
         finally:
@@ -1677,7 +1677,7 @@ async def _send_handling_redirects(
                     response.next_request = request
                     return response
 
-            except Exception as exc:
+            except BaseException as exc:
                 await response.aclose()
                 raise exc
 
diff --git a/tests/client/test_async_client.py b/tests/client/test_async_client.py
index 219d612f79..a0ceea22a3 100644
--- a/tests/client/test_async_client.py
+++ b/tests/client/test_async_client.py
@@ -1,3 +1,4 @@
+import asyncio
 import typing
 from datetime import timedelta
 
@@ -331,3 +332,16 @@ async def test_server_extensions(server):
         response = await client.get(url)
     assert response.status_code == 200
     assert response.extensions["http_version"] == b"HTTP/1.1"
+
+
+@pytest.mark.asyncio
+async def test_cancelled_response(server):
+    async with httpx.AsyncClient() as client:
+        url = server.url.join("/drip?delay=0&duration=0.1")
+        response = await asyncio.wait_for(client.get(url), 0.2)
+        assert response.status_code == 200
+        assert response.content == b"*"
+    async with httpx.AsyncClient() as client:
+        url = server.url.join("/drip?delay=0&duration=0.5")
+        with pytest.raises(asyncio.TimeoutError):
+            await asyncio.wait_for(client.get(url), 0.2)
diff --git a/tests/conftest.py b/tests/conftest.py
index 970c353547..877afe9159 100644
--- a/tests/conftest.py
+++ b/tests/conftest.py
@@ -4,6 +4,7 @@
 import threading
 import time
 import typing
+import urllib.parse
 
 import pytest
 import trustme
@@ -76,6 +77,8 @@ async def app(scope, receive, send):
     assert scope["type"] == "http"
     if scope["path"].startswith("/slow_response"):
         await slow_response(scope, receive, send)
+    if scope["path"].startswith("/drip"):
+        await drip_response(scope, receive, send)
     elif scope["path"].startswith("/status"):
         await status_code(scope, receive, send)
     elif scope["path"].startswith("/echo_body"):
@@ -126,6 +129,29 @@ async def slow_response(scope, receive, send):
     await send({"type": "http.response.body", "body": b"Hello, world!"})
 
 
+async def drip_response(scope, receive, send):
+    """
+    Drips data over a duration after an optional initial delay.
+    eg: https://httpbin.org/drip?delay=0&duration=1
+    """
+    qs = urllib.parse.parse_qs(scope["query_string"].decode())
+    delay = float(qs.get("delay", ["0"])[0])
+    duration = float(qs.get("duration", ["1"])[0])
+    await sleep(delay)
+    await send(
+        {
+            "type": "http.response.start",
+            "status": 200,
+            "headers": [[b"content-type", b"text/plain"]],
+        }
+    )
+    drip = {"type": "http.response.body", "body": b"*", "more_body": True}
+    for _ in range(int(duration * 10)):
+        await send(drip)
+        await sleep(0.1)
+    await send({"type": "http.response.body", "body": b""})
+
+
 async def status_code(scope, receive, send):
     status_code = int(scope["path"].replace("/status/", ""))
     await send(