If you have an ASP.NET Web Site project type (the one without a proper .csproj or .vpproj project file and that is just a folder of loose files), then when you add a package
-- EDIT -- See updated version of this solution above.
Just add a class library project to the solution, and add a reference to the class library project to your website.
Then, add the nuget in the class library, not the website project.
When your website builds, it pulls in all the class libraries' dependencies in turn.