Another MVC framework I’ve used – Symfony2 – uses a dependency injector to carefully configure and manage dependencies.
Does ASP.NET MVC have anything similar which would let me configure a MongoDB connection and then pass it down to my controller logic where it can be used?
There are many different dependency injection libraries you can use with Asp.net MVC, here’s a (non comprehensive) list:
If you go looking for them in Nuget, typically there will be one package for the container itself, and a different package that adds the Asp.Net MVC plugins.
Take a look at them, figure out which flavor you like, then configure your mongo db connection to be per-request, add the connection to your controller constructor, and away you go. If you pick a lib and update your question, I’m sure someone will answer with specific code etc.