I have found a few questions (this one, this one, and this one) related to this question, however none of them clearly answer my question: Do I really need TFS or VSO in order t
The answer is quite simple. Just bind the two Web Apps (one for the demo.Web and the other for demo.Api) to the same repo. Then, go to each other's Settings blade in the Azure Portal and look for Application Settings.
On the App Settings subsection, add a setting called "Project" whose value is the path to the .csproj that you want to build, relative to the repo's root folder.
On the first Web App it will point to the demo.Web project file, and on the second Web App, it will point to the demo.Api project file.
You don't need TFS to achieve it.
EDIT: For ASPNET5/MVC6 projects, just add the path to the "project.json" folder you want to deploy, you don't need to add the .xproj to the path.