Of course, you will not only perform unit tests or API tests. You also need to test your UI, if you have one.

I'd recommend to do it using the Selenium Webdriver. You can find more information here, http://www.seleniumhq.org/docs/03_webdriver.jsp.

In this case, you would probably want to deploy your application to a QA environment, run the UI tests, and then go into production if everything is fine. So it's just a matter of adding some new jobs to your pipeline, now you know how to do it.