diff --git a/lib/_stream_readable.js b/lib/_stream_readable.js
index 0d83ada205b835..8f5f66b391293c 100644
--- a/lib/_stream_readable.js
+++ b/lib/_stream_readable.js
@@ -190,7 +190,7 @@ function Readable(options) {
       this._destroy = options.destroy;
   }
 
-  Stream.call(this);
+  Stream.call(this, options);
 }
 
 ObjectDefineProperty(Readable.prototype, 'destroyed', {
@@ -233,6 +233,14 @@ Readable.prototype._destroy = function(err, cb) {
   cb(err);
 };
 
+Readable.prototype[EE.captureRejectionSymbol] = function(err) {
+  // TODO(mcollina): remove the destroyed if once errorEmitted lands in
+  // Readable.
+  if (!this.destroyed) {
+    this.destroy(err);
+  }
+};
+
 // Manually shove something into the read() buffer.
 // This returns true if the highWaterMark has not been hit yet,
 // similar to how Writable.write() returns true if you should
diff --git a/lib/_stream_writable.js b/lib/_stream_writable.js
index 9bea209e257d2a..3452ac810beec9 100644
--- a/lib/_stream_writable.js
+++ b/lib/_stream_writable.js
@@ -35,6 +35,7 @@ module.exports = Writable;
 Writable.WritableState = WritableState;
 
 const internalUtil = require('internal/util');
+const EE = require('events');
 const Stream = require('stream');
 const { Buffer } = require('buffer');
 const destroyImpl = require('internal/streams/destroy');
@@ -254,7 +255,7 @@ function Writable(options) {
       this._final = options.final;
   }
 
-  Stream.call(this);
+  Stream.call(this, options);
 }
 
 // Otherwise people can pipe Writable streams, which is just wrong.
@@ -804,3 +805,7 @@ Writable.prototype._undestroy = destroyImpl.undestroy;
 Writable.prototype._destroy = function(err, cb) {
   cb(err);
 };
+
+Writable.prototype[EE.captureRejectionSymbol] = function(err) {
+  this.destroy(err);
+};
diff --git a/lib/internal/streams/legacy.js b/lib/internal/streams/legacy.js
index 702e3c56ba6376..2bc7a86aa050b6 100644
--- a/lib/internal/streams/legacy.js
+++ b/lib/internal/streams/legacy.js
@@ -6,8 +6,8 @@ const {
 
 const EE = require('events');
 
-function Stream() {
-  EE.call(this);
+function Stream(opts) {
+  EE.call(this, opts);
 }
 ObjectSetPrototypeOf(Stream.prototype, EE.prototype);
 ObjectSetPrototypeOf(Stream, EE);
diff --git a/test/parallel/test-stream-catch-rejections.js b/test/parallel/test-stream-catch-rejections.js
new file mode 100644
index 00000000000000..fb5f1fccc18bd2
--- /dev/null
+++ b/test/parallel/test-stream-catch-rejections.js
@@ -0,0 +1,52 @@
+'use strict';
+
+const common = require('../common');
+const stream = require('stream');
+const assert = require('assert');
+
+{
+  const r = new stream.Readable({
+    captureRejections: true,
+    read() {
+      this.push('hello');
+      this.push('world');
+      this.push(null);
+    }
+  });
+
+  const err = new Error('kaboom');
+
+  r.on('error', common.mustCall((_err) => {
+    assert.strictEqual(err, _err);
+    assert.strictEqual(r.destroyed, true);
+  }));
+
+  r.on('data', async () => {
+    throw err;
+  });
+}
+
+{
+  const w = new stream.Writable({
+    captureRejections: true,
+    highWaterMark: 1,
+    write(chunk, enc, cb) {
+      cb();
+    }
+  });
+
+  const err = new Error('kaboom');
+
+  w.write('hello', () => {
+    w.write('world');
+  });
+
+  w.on('error', common.mustCall((_err) => {
+    assert.strictEqual(err, _err);
+    assert.strictEqual(w.destroyed, true);
+  }));
+
+  w.on('drain', common.mustCall(async () => {
+    throw err;
+  }, 2));
+}