diff --git a/Lib/asyncio/staggered.py b/Lib/asyncio/staggered.py
index c3a7441a7b091d..83871f25297660 100644
--- a/Lib/asyncio/staggered.py
+++ b/Lib/asyncio/staggered.py
@@ -68,6 +68,7 @@ async def staggered_race(coro_fns, delay, *, loop=None):
     winner_index = None
     exceptions = []
     running_tasks = []
+    waiting_tasks = []
 
     async def run_one_coro(previous_failed) -> None:
         # Wait for the previous task to finish, or for delay seconds
@@ -78,6 +79,7 @@ async def run_one_coro(previous_failed) -> None:
                 # cancelled, otherwise there will be a "Task destroyed but it is
                 # pending" later.
                 await tasks.wait_for(previous_failed.wait(), delay)
+                waiting_tasks.append(previous_failed)
         # Get the next coroutine to run
         try:
             this_index, coro_fn = next(enum_coro_fns)
@@ -87,7 +89,11 @@ async def run_one_coro(previous_failed) -> None:
         this_failed = locks.Event()
         next_task = loop.create_task(run_one_coro(this_failed))
         running_tasks.append(next_task)
-        assert len(running_tasks) == this_index + 2
+        running_tasks.append(next_task)
+        if len(waiting_tasks) != len(running_tasks):
+            task = running_tasks.pop(len(waiting_tasks) - 1)
+            if task.cancelled():
+                task.cancel()
         # Prepare place to put this coroutine's exceptions if not won
         exceptions.append(None)
         assert len(exceptions) == this_index + 1