diff --git a/Lib/asyncio/tasks.py b/Lib/asyncio/tasks.py
index 9a9d0d6e3cc269..d0d8c80a5b9324 100644
--- a/Lib/asyncio/tasks.py
+++ b/Lib/asyncio/tasks.py
@@ -407,9 +407,9 @@ async def wait_for(fut, timeout):
     if timeout is None:
         return await fut
 
-    if timeout <= 0:
-        fut = ensure_future(fut, loop=loop)
+    fut = ensure_future(fut, loop=loop)
 
+    if timeout <= 0:
         if fut.done():
             return fut.result()
 
@@ -421,47 +421,24 @@ async def wait_for(fut, timeout):
         else:
             raise exceptions.TimeoutError()
 
-    waiter = loop.create_future()
-    timeout_handle = loop.call_later(timeout, _release_waiter, waiter)
-    cb = functools.partial(_release_waiter, waiter)
+    timeout_occurred = False
 
-    fut = ensure_future(fut, loop=loop)
-    fut.add_done_callback(cb)
+    def _on_timeout(fut):
+        nonlocal timeout_occurred
+        timeout_occurred = fut.cancel()
 
+    timeout_handle = loop.call_later(timeout, _on_timeout, fut)
+    cb = fut.add_done_callback(lambda fut: timeout_handle.cancel())
     try:
-        # wait until the future completes or the timeout
-        try:
-            await waiter
-        except exceptions.CancelledError:
-            if fut.done():
-                return fut.result()
-            else:
-                fut.remove_done_callback(cb)
-                # We must ensure that the task is not running
-                # after wait_for() returns.
-                # See https://bugs.python.org/issue32751
-                await _cancel_and_wait(fut, loop=loop)
-                raise
-
-        if fut.done():
-            return fut.result()
+        return await fut
+    except exceptions.CancelledError as exc:
+        if timeout_occurred:
+            raise exceptions.TimeoutError() from exc
         else:
-            fut.remove_done_callback(cb)
-            # We must ensure that the task is not running
-            # after wait_for() returns.
-            # See https://bugs.python.org/issue32751
-            await _cancel_and_wait(fut, loop=loop)
-            # In case task cancellation failed with some
-            # exception, we should re-raise it
-            # See https://bugs.python.org/issue40607
-            try:
-                fut.result()
-            except exceptions.CancelledError as exc:
-                raise exceptions.TimeoutError() from exc
-            else:
-                raise exceptions.TimeoutError()
+            raise
     finally:
         timeout_handle.cancel()
+        fut.remove_done_callback(cb)
 
 
 async def _wait(fs, timeout, return_when, loop):