diff --git a/lib/internal/http2/core.js b/lib/internal/http2/core.js
index cb9bfc37ce2cd7..0ea3ca75f4d43e 100644
--- a/lib/internal/http2/core.js
+++ b/lib/internal/http2/core.js
@@ -20,8 +20,9 @@ const { onServerStream,
         Http2ServerResponse,
 } = require('internal/http2/compat');
 const { utcDate } = require('internal/http');
+const { promisify } = require('internal/util');
 const { _connectionListener: httpConnectionListener } = require('http');
-const { isUint8Array } = process.binding('util');
+const { isUint8Array, createPromise, promiseResolve } = process.binding('util');
 const debug = util.debuglog('http2');
 
 
@@ -2454,6 +2455,17 @@ function connect(authority, options, listener) {
   return session;
 }
 
+// Support util.promisify
+Object.defineProperty(connect, promisify.custom, {
+  value: (authority, options) => {
+    const promise = createPromise();
+    const server = connect(authority,
+                           options,
+                           () => promiseResolve(promise, server));
+    return promise;
+  }
+});
+
 function createSecureServer(options, handler) {
   if (options == null || typeof options !== 'object') {
     throw new errors.TypeError('ERR_INVALID_ARG_TYPE',
diff --git a/src/node_http2_core-inl.h b/src/node_http2_core-inl.h
index c6654123dda656..0d2a697a9d73e0 100644
--- a/src/node_http2_core-inl.h
+++ b/src/node_http2_core-inl.h
@@ -52,17 +52,19 @@ inline int Nghttp2Session::OnBeginHeadersCallback(nghttp2_session* session,
                                                   const nghttp2_frame* frame,
                                                   void* user_data) {
   Nghttp2Session* handle = static_cast<Nghttp2Session*>(user_data);
+  // If this is a push promise frame, we want to grab the handle of
+  // the promised stream.
   int32_t id = (frame->hd.type == NGHTTP2_PUSH_PROMISE) ?
-  frame->push_promise.promised_stream_id :
-  frame->hd.stream_id;
+      frame->push_promise.promised_stream_id :
+      frame->hd.stream_id;
   DEBUG_HTTP2("Nghttp2Session %s: beginning headers for stream %d\n",
               handle->TypeName(handle->type()), id);
 
   Nghttp2Stream* stream = handle->FindStream(id);
   if (stream == nullptr) {
-  Nghttp2Stream::Init(id, handle, frame->headers.cat);
+    Nghttp2Stream::Init(id, handle, frame->headers.cat);
   } else {
-  stream->StartHeaders(frame->headers.cat);
+    stream->StartHeaders(frame->headers.cat);
   }
   return 0;
 }
@@ -77,9 +79,11 @@ inline int Nghttp2Session::OnHeaderCallback(nghttp2_session* session,
                                             uint8_t flags,
                                             void* user_data) {
   Nghttp2Session* handle = static_cast<Nghttp2Session*>(user_data);
+  // If this is a push promise frame, we want to grab the handle of
+  // the promised stream.
   int32_t id = (frame->hd.type == NGHTTP2_PUSH_PROMISE) ?
-  frame->push_promise.promised_stream_id :
-  frame->hd.stream_id;
+      frame->push_promise.promised_stream_id :
+      frame->hd.stream_id;
   Nghttp2Stream* stream = handle->FindStream(id);
   nghttp2_header_list* header = header_free_list.pop();
   header->name = name;
diff --git a/src/node_http2_core.h b/src/node_http2_core.h
index 2cd669f118a403..69b9891d5871cb 100644
--- a/src/node_http2_core.h
+++ b/src/node_http2_core.h
@@ -141,9 +141,9 @@ class Nghttp2Session {
   inline ssize_t Write(const uv_buf_t* bufs, unsigned int nbufs);
 
   // Returns the nghttp2 library session
-  inline nghttp2_session* session() { return session_; }
+  inline nghttp2_session* session() const { return session_; }
 
-  nghttp2_session_type type() {
+  nghttp2_session_type type() const {
     return session_type_;
   }
 
diff --git a/test/parallel/test-http2-client-promisify-connect.js b/test/parallel/test-http2-client-promisify-connect.js
new file mode 100644
index 00000000000000..5b6dccad6a223e
--- /dev/null
+++ b/test/parallel/test-http2-client-promisify-connect.js
@@ -0,0 +1,34 @@
+// Flags: --expose-http2
+'use strict';
+
+const common = require('../common');
+if (!common.hasCrypto)
+  common.skip('missing crypto');
+const assert = require('assert');
+const http2 = require('http2');
+const util = require('util');
+
+const server = http2.createServer();
+server.on('stream', common.mustCall((stream) => {
+  stream.respond();
+  stream.end('ok');
+}));
+server.listen(0, common.mustCall(() => {
+
+  const connect = util.promisify(http2.connect);
+
+  connect(`http://localhost:${server.address().port}`)
+    .catch(common.mustNotCall())
+    .then(common.mustCall((client) => {
+      assert(client);
+      const req = client.request();
+      let data = '';
+      req.setEncoding('utf8');
+      req.on('data', (chunk) => data += chunk);
+      req.on('end', common.mustCall(() => {
+        assert.strictEqual(data, 'ok');
+        client.destroy();
+        server.close();
+      }));
+    }));
+}));