You could also use YouTube Vanced. YouTube Vanced is a modded version of YouTube. It is a feature enhanced version of the official YouTube app, most importantly brings the feature for background playback (as well as other equally useful features including built-in adblocking, black/dark themes, etc).
It officially supports minimum (API 17) version 4.2 jellybean and works on both rooted and non-rooted devices.
This page has installation instructions and download links for non-rooted and rooted devices.
Additionally you need to install the install MicroG package (found in the download links) in order to be able to log into google account (not necessary if you don't want to log in)
(Click image to enlarge)