diff --git a/lib/_stream_readable.js b/lib/_stream_readable.js
index 2a2abd4715..a31e5d0a66 100644
--- a/lib/_stream_readable.js
+++ b/lib/_stream_readable.js
@@ -305,12 +305,14 @@ Readable.prototype.pipe = function(dest, pipeOpts) {
       dest !== process.stdout &&
       dest !== process.stderr) {
     src.once('end', onend);
-    dest.on('unpipe', function unpipe(readable) {
-      if (readable === src) {
-        src.removeListener('end', onend);
-        dest.removeListener('unpipe', unpipe);
-      }
-    });
+  } else {
+    src.once('end', cleanup);
+  }
+
+  dest.on('unpipe', onunpipe);
+  function onunpipe(readable) {
+    if (readable !== src) return;
+    cleanup();
   }
 
   if (pipeOpts && pipeOpts.chunkSize)
@@ -326,19 +328,25 @@ Readable.prototype.pipe = function(dest, pipeOpts) {
   // too slow.
   var ondrain = pipeOnDrain(src);
   dest.on('drain', ondrain);
-  dest.on('unpipe', function cleanup(readable) {
-    if (readable === src) {
-      dest.removeListener('unpipe', cleanup);
-
-      // if the reader is waiting for a drain event from this
-      // specific writer, then it would cause it to never start
-      // flowing again.
-      // So, if this is awaiting a drain, then we just call it now.
-      // If we don't know, then assume that we are waiting for one.
-      if (!dest._writableState || dest._writableState.needDrain)
-        ondrain();
-    }
-  });
+
+  function cleanup() {
+    // cleanup event handlers once the pipe is broken
+    dest.removeListener('close', unpipe);
+    dest.removeListener('finish', onfinish);
+    dest.removeListener('drain', ondrain);
+    dest.removeListener('error', onerror);
+    dest.removeListener('unpipe', onunpipe);
+    src.removeListener('end', onend);
+    src.removeListener('end', cleanup);
+
+    // if the reader is waiting for a drain event from this
+    // specific writer, then it would cause it to never start
+    // flowing again.
+    // So, if this is awaiting a drain, then we just call it now.
+    // If we don't know, then assume that we are waiting for one.
+    if (!dest._writableState || dest._writableState.needDrain)
+      ondrain();
+  }
 
   // if the dest has an error, then stop piping into it.
   // however, don't suppress the throwing behavior for this.
@@ -361,17 +369,6 @@ Readable.prototype.pipe = function(dest, pipeOpts) {
     src.unpipe(dest);
   }
 
-  // cleanup event handlers once the pipe is broken
-  dest.once('unpipe', function cleanup(readable) {
-    if (readable !== src) return;
-
-    dest.removeListener('close', unpipe);
-    dest.removeListener('finish', onfinish);
-    dest.removeListener('drain', ondrain);
-    dest.removeListener('error', onerror);
-    dest.removeListener('unpipe', cleanup);
-  });
-
   // tell the dest that it's being piped to
   dest.emit('pipe', src);