diff --git a/src/openai/_streaming.py b/src/openai/_streaming.py
index 45c13cc11d..006ab34f21 100644
--- a/src/openai/_streaming.py
+++ b/src/openai/_streaming.py
@@ -61,6 +61,14 @@ def __stream__(self) -> Iterator[_T]:
         try:
             for sse in iterator:
                 if sse.data.startswith("[DONE]"):
+                    # Drain remaining bytes so h11 fully parses the chunked terminator
+                    # before close(), allowing the connection to be returned to the pool.
+                    # Errors are suppressed — the stream is done at [DONE] regardless.
+                    try:
+                        for _ in iterator:
+                            pass
+                    except Exception:
+                        pass
                     break
 
                 # we have to special case the Assistants `thread.` events since we won't have an "event" key in the data
@@ -171,6 +179,14 @@ async def __stream__(self) -> AsyncIterator[_T]:
         try:
             async for sse in iterator:
                 if sse.data.startswith("[DONE]"):
+                    # Drain remaining bytes so h11 fully parses the chunked terminator
+                    # before aclose(), allowing the connection to be returned to the pool.
+                    # Errors are suppressed — the stream is done at [DONE] regardless.
+                    try:
+                        async for _ in iterator:
+                            pass
+                    except Exception:
+                        pass
                     break
 
                 # we have to special case the Assistants `thread.` events since we won't have an "event" key in the data