diff --git a/cleo/application.py b/cleo/application.py
index 44a79709..9b17d1a1 100644
--- a/cleo/application.py
+++ b/cleo/application.py
@@ -325,6 +325,12 @@ def run(
 
             try:
                 exit_code = self._run(io)
+            except BrokenPipeError:
+                # If we are piped to another process, it may close early and send a
+                # SIGPIPE: https://docs.python.org/3/library/signal.html#note-on-sigpipe
+                devnull = os.open(os.devnull, os.O_WRONLY)
+                os.dup2(devnull, sys.stdout.fileno())
+                exit_code = 0
             except Exception as e:
                 if not self._catch_exceptions:
                     raise
diff --git a/cleo/io/outputs/stream_output.py b/cleo/io/outputs/stream_output.py
index 71d16aa6..a5140028 100644
--- a/cleo/io/outputs/stream_output.py
+++ b/cleo/io/outputs/stream_output.py
@@ -77,6 +77,7 @@ def _write(self, message: str, new_line: bool = False) -> None:
             message += "\n"
 
         self._stream.write(message)
+        self._stream.flush()
 
     def _has_color_support(self) -> bool:
         # Follow https://no-color.org/