The steps I needed to perform were:
- Add reference to System.Web.Http.WebHost.
- Add App_Start\WebApiConfig.cs(see code snippet below).
- Import namespace System.Web.HttpinGlobal.asax.cs.
- Call WebApiConfig.Register(GlobalConfiguration.Configuration)inMvcApplication.Application_Start()(in fileGlobal.asax.cs), before registering the default Web Application route as that would otherwise take precedence.
- Add a controller deriving from System.Web.Http.ApiController.
I could then learn enough from the tutorial (Your First ASP.NET Web API) to define my API controller.
App_Start\WebApiConfig.cs:
using System.Web.Http;
class WebApiConfig
{
    public static void Register(HttpConfiguration configuration)
    {
        configuration.Routes.MapHttpRoute("API Default", "api/{controller}/{id}",
            new { id = RouteParameter.Optional });
    }
}
Global.asax.cs:
using System.Web.Http;
...
protected void Application_Start()
{
    AreaRegistration.RegisterAllAreas();
    RegisterGlobalFilters(GlobalFilters.Filters);
    WebApiConfig.Register(GlobalConfiguration.Configuration);
    RegisterRoutes(RouteTable.Routes);
    BundleConfig.RegisterBundles(BundleTable.Bundles);
}
Update 10.16.2015:
Word has it, the NuGet package Microsoft.AspNet.WebApi must be installed for the above to work.