diff --git a/tokio-postgres/src/client.rs b/tokio-postgres/src/client.rs
index 8a7c9d288..2ab5c53b3 100644
--- a/tokio-postgres/src/client.rs
+++ b/tokio-postgres/src/client.rs
@@ -21,12 +21,13 @@ use crate::{
 use bytes::{Buf, BytesMut};
 use fallible_iterator::FallibleIterator;
 use futures_channel::mpsc;
-use futures_util::{future, pin_mut, ready, StreamExt, TryStreamExt};
+use futures_util::{future, pin_mut, ready, Stream, StreamExt, TryStreamExt};
 use parking_lot::Mutex;
 use postgres_protocol::message::{backend::Message, frontend};
 use postgres_types::BorrowToSql;
 use std::collections::HashMap;
 use std::fmt;
+use std::pin::Pin;
 use std::sync::Arc;
 use std::task::{Context, Poll};
 #[cfg(feature = "runtime")]
@@ -59,6 +60,17 @@ impl Responses {
     }
 }
 
+impl Stream for Responses {
+    type Item = Result<Message, Error>;
+
+    fn poll_next(mut self: Pin<&mut Self>, cx: &mut Context<'_>) -> Poll<Option<Self::Item>> {
+        match ready!((*self).poll_next(cx)) {
+            Err(err) if err.is_closed() => Poll::Ready(None),
+            msg => Poll::Ready(Some(msg)),
+        }
+    }
+}
+
 /// A cache of type info and prepared statements for fetching type info
 /// (corresponding to the queries in the [prepare](prepare) module).
 #[derive(Default)]