There's a super easy way actually. This scenario was anticipated by the Azure team and there's a good way to handle it. It's simple too.
You simply create a text file at the root of your project called .deployment
. In the .deployment file you add the following text...
[config]
project = mysubfolder
When you either Git deploy or use CI to deploy from source control, the entire repository is deployed, but the .deployment file tells Kudu (that's the engine that handles your website management) where the actual website (or node project) is.
You can find more info here.
Also, check out this post where I mention an alternative strategy for project repos in case that helps.