So you want the WebAPI to only be used by the MVC page? The best architectural method is to separate the two rather than leave both in one project. Why? Because the MVC app is a experience layer for humans. The WebAPI is an experience layer for the MVC app. Move it back where it can't be accessed.
You can add on tokens, etc, but the MVC app sits on the server, but is accessed on the client computer. The wider the scope of the application (ie, intranet or internet or something in between?), the more difficult the problem and the harder it is for your users to access the application. Moving the WebAPI internal and leaving the MVC app exposed guarantees external users cannot use the API.
The main reason WebAPI and MVC exist together in a single project (still a mistake in most instances, IMO) is you are exposing both to the same audience. If that is not your intent, don't do it.