diff --git a/Lib/asyncio/timeouts.py b/Lib/asyncio/timeouts.py
index 09342dc7c1310b..12b11b84193b27 100644
--- a/Lib/asyncio/timeouts.py
+++ b/Lib/asyncio/timeouts.py
@@ -89,6 +89,7 @@ async def __aenter__(self) -> "Timeout":
         self._state = _State.ENTERED
         self._task = task
         self._cancelling = self._task.cancelling()
+        self._must_cancel = self._task._must_cancel
         self.reschedule(self._when)
         return self
 
@@ -106,10 +107,13 @@ async def __aexit__(
 
         if self._state is _State.EXPIRING:
             self._state = _State.EXPIRED
-
-            if self._task.uncancel() <= self._cancelling and exc_type is not None:
-                # Since there are no new cancel requests, we're
-                # handling this.
+            if (
+                self._task.uncancel() <= self._cancelling
+                and exc_type is not None
+                and not self._must_cancel
+            ):
+                # Since there are no new cancel requests
+                # and the task doesn't _have to_ raise CancelledError, we're handling this.
                 if issubclass(exc_type, exceptions.CancelledError):
                     raise TimeoutError from exc_val
                 elif exc_val is not None:
diff --git a/Lib/test/test_asyncio/test_timeouts.py b/Lib/test/test_asyncio/test_timeouts.py
index 3ba84d63b2ca5f..bb9c9a39c9f883 100644
--- a/Lib/test/test_asyncio/test_timeouts.py
+++ b/Lib/test/test_asyncio/test_timeouts.py
@@ -4,6 +4,7 @@
 import time
 
 import asyncio
+from asyncio import CancelledError
 
 from test.test_asyncio.utils import await_without_task
 
@@ -298,6 +299,18 @@ async def test_nested_timeout_in_finally(self):
         self.assertIs(e2.__context__, e22)
 
     async def test_timeout_after_cancellation(self):
+        asyncio.current_task().cancel()
+        with self.assertRaises(asyncio.CancelledError):
+            async with asyncio.timeout(0.1):
+                await asyncio.sleep(1)
+
+    async def test_timeout_immediately_after_cancellation(self):
+        asyncio.current_task().cancel()
+        with self.assertRaises(asyncio.CancelledError):
+            async with asyncio.timeout(0.0):
+                await asyncio.sleep(1)
+
+    async def test_timeout_while_handling_cancellation(self):
         try:
             asyncio.current_task().cancel()
             await asyncio.sleep(1)  # work which will be cancelled
@@ -308,7 +321,7 @@ async def test_timeout_after_cancellation(self):
                 async with asyncio.timeout(0.0):
                     await asyncio.sleep(1)  # some cleanup
 
-    async def test_cancel_in_timeout_after_cancellation(self):
+    async def test_cancel_in_timeout_while_handling_cancellation(self):
         try:
             asyncio.current_task().cancel()
             await asyncio.sleep(1)  # work which will be cancelled