This is actually easier than it first appears though I've not tried it in a long time.
The Azure web service (or whatever they are calling it this week) includes the ability to run PHP, Python (I think) and Node.JS
The Node.js integration uses an IIS plugin (IIS is Microsoft's web server rather than Apache or NGINX). So you can run Node-RED fairly easily though you probably need to install Node-RED locally rather than globally. I have some instructions for doing a local Node-RED install on my GitHub site somewhere - in one of my repo's. That is what I always use anyway as I find it a lot more convenient, flexible and secure.
Of course, you will need to secure Node-RED well since it will be exposed to the Internet if you use the Azure web service. If you need more security, you will need to look at the other Azure features. You might, for example, restrict access to the admin UI to only certain source IP address ranges.
You can also add Azure Active Directory authentication to the whole thing which would be very much recommended. Again, this is native to Azure, you wouldn't need changes in Node-RED.