diff --git a/doc/api/errors.md b/doc/api/errors.md
index 918cfa343136af..1e091b46a1d8d3 100644
--- a/doc/api/errors.md
+++ b/doc/api/errors.md
@@ -1353,7 +1353,14 @@ An attempt was made to `require()` an [ES6 module][].
 
 The [`server.listen()`][] method was called while a `net.Server` was already
 listening. This applies to all instances of `net.Server`, including HTTP, HTTPS,
-and HTTP/2 Server instances.
+and HTTP/2 `Server` instances.
+
+<a id="ERR_SERVER_NOT_RUNNING"></a>
+### ERR_SERVER_NOT_RUNNING
+
+The [`server.close()`][] method was called when a `net.Server` was not
+running. This applies to all instances of `net.Server`, including HTTP, HTTPS,
+and HTTP/2 `Server` instances.
 
 <a id="ERR_SOCKET_ALREADY_BOUND"></a>
 ### ERR_SOCKET_ALREADY_BOUND
diff --git a/lib/internal/errors.js b/lib/internal/errors.js
index 7276bb6344a5bd..72db142c7a0e81 100644
--- a/lib/internal/errors.js
+++ b/lib/internal/errors.js
@@ -447,6 +447,7 @@ E('ERR_PARSE_HISTORY_DATA', 'Could not parse history data in %s');
 E('ERR_REQUIRE_ESM', 'Must use import to load ES Module: %s');
 E('ERR_SERVER_ALREADY_LISTEN',
   'Listen method has been called more than once without closing.');
+E('ERR_SERVER_NOT_RUNNING', 'Server is not running.');
 E('ERR_SOCKET_ALREADY_BOUND', 'Socket is already bound');
 E('ERR_SOCKET_BAD_BUFFER_SIZE', 'Buffer size must be a positive integer');
 E('ERR_SOCKET_BAD_PORT', 'Port should be > 0 and < 65536. Received %s.');
diff --git a/lib/net.js b/lib/net.js
index 245415d87c63a1..bfafbcc21915ba 100644
--- a/lib/net.js
+++ b/lib/net.js
@@ -724,7 +724,7 @@ Socket.prototype._writeGeneric = function(writev, data, encoding, cb) {
   this._unrefTimer();
 
   if (!this._handle) {
-    this.destroy(new Error('This socket is closed'), cb);
+    this.destroy(new errors.Error('ERR_SOCKET_CLOSED'), cb);
     return false;
   }
 
@@ -1624,7 +1624,7 @@ Server.prototype.close = function(cb) {
   if (typeof cb === 'function') {
     if (!this._handle) {
       this.once('close', function close() {
-        cb(new Error('Not running'));
+        cb(new errors.Error('ERR_SERVER_NOT_RUNNING'));
       });
     } else {
       this.once('close', cb);
diff --git a/test/parallel/test-http-unix-socket.js b/test/parallel/test-http-unix-socket.js
index 6d5897cacbecc2..7a17b9bc9ca50a 100644
--- a/test/parallel/test-http-unix-socket.js
+++ b/test/parallel/test-http-unix-socket.js
@@ -58,8 +58,10 @@ server.listen(common.PIPE, common.mustCall(function() {
       assert.strictEqual(res.body, 'hello world\n');
       server.close(common.mustCall(function(error) {
         assert.strictEqual(error, undefined);
-        server.close(common.mustCall(function(error) {
-          assert.strictEqual(error && error.message, 'Not running');
+        server.close(common.expectsError({
+          code: 'ERR_SERVER_NOT_RUNNING',
+          message: 'Server is not running.',
+          type: Error
         }));
       }));
     }));
diff --git a/test/parallel/test-net-socket-destroy-send.js b/test/parallel/test-net-socket-destroy-send.js
index 6dc4b9566a3900..a602b89253887d 100644
--- a/test/parallel/test-net-socket-destroy-send.js
+++ b/test/parallel/test-net-socket-destroy-send.js
@@ -12,11 +12,16 @@ server.listen(0, common.mustCall(function() {
   conn.on('connect', common.mustCall(function() {
     // Test destroy returns this, even on multiple calls when it short-circuits.
     assert.strictEqual(conn, conn.destroy().destroy());
-    conn.on('error', common.mustCall(function(err) {
-      assert.strictEqual(err.message, 'This socket is closed');
+    conn.on('error', common.expectsError({
+      code: 'ERR_SOCKET_CLOSED',
+      message: 'Socket is closed',
+      type: Error
     }));
-    conn.write(Buffer.from('kaboom'), common.mustCall(function(err) {
-      assert.strictEqual(err.message, 'This socket is closed');
+
+    conn.write(Buffer.from('kaboom'), common.expectsError({
+      code: 'ERR_SOCKET_CLOSED',
+      message: 'Socket is closed',
+      type: Error
     }));
     server.close();
   }));
diff --git a/test/parallel/test-net-socket-write-after-close.js b/test/parallel/test-net-socket-write-after-close.js
index d3a3d937d338ab..e3593a3a09a146 100644
--- a/test/parallel/test-net-socket-write-after-close.js
+++ b/test/parallel/test-net-socket-write-after-close.js
@@ -26,11 +26,14 @@ const net = require('net');
   server.listen(common.mustCall(() => {
     const port = server.address().port;
     const client = net.connect({ port }, common.mustCall(() => {
-      client.on('error', common.mustCall((err) => {
-        server.close();
-        assert.strictEqual(err.constructor, Error);
-        assert.strictEqual(err.message, 'This socket is closed');
+      client.on('error', common.expectsError({
+        code: 'ERR_SOCKET_CLOSED',
+        message: 'Socket is closed',
+        type: Error
       }));
+
+      server.close();
+
       client._handle.close();
       client._handle = null;
       client.write('foo');