After struggling for a whole evening I finally got this to work. After some debugging I found the problem I was walking into was that my client was sending a so called preflight Options request to check if the application was allowed to send a post request with the origin, methods and headers provided. I didn't want to use Owin or an APIController, so I started digging and came up with the following solution with just an ActionFilterAttribute. Especially the "Access-Control-Allow-Headers" part is very important, as the headers mentioned there do have to match the headers your request will send.
using System.Linq;
using System.Web;
using System.Web.Mvc;
namespace MyNamespace
{
    public class AllowCrossSiteJsonAttribute : ActionFilterAttribute
    {
        public override void OnActionExecuting(ActionExecutingContext filterContext)
        {
            HttpRequest request = HttpContext.Current.Request;
            HttpResponse response = HttpContext.Current.Response;
            // check for preflight request
            if (request.Headers.AllKeys.Contains("Origin") && request.HttpMethod == "OPTIONS")
            {
                response.AppendHeader("Access-Control-Allow-Origin", "*");
                response.AppendHeader("Access-Control-Allow-Credentials", "true");
                response.AppendHeader("Access-Control-Allow-Methods", "GET, PUT, POST, DELETE");
                response.AppendHeader("Access-Control-Allow-Headers", "Origin, X-Requested-With, X-RequestDigest, Cache-Control, Content-Type, Accept, Access-Control-Allow-Origin, Session, odata-version");
                response.End();
            }
            else
            {
                HttpContext.Current.Response.Cache.SetCacheability(HttpCacheability.NoCache);
                HttpContext.Current.Response.Cache.SetNoStore();
                response.AppendHeader("Access-Control-Allow-Origin", "*");
                response.AppendHeader("Access-Control-Allow-Credentials", "true");
                if (request.HttpMethod == "POST")
                {
                    response.AppendHeader("Access-Control-Allow-Methods", "GET, PUT, POST, DELETE");
                    response.AppendHeader("Access-Control-Allow-Headers", "Origin, X-Requested-With, X-RequestDigest, Cache-Control, Content-Type, Accept, Access-Control-Allow-Origin, Session, odata-version");
                }
                base.OnActionExecuting(filterContext);
            }
        }
    }
}
Finally, my MVC action method looks like this. Important here is to also mention the Options HttpVerbs, because otherwise the preflight request will fail.
[AcceptVerbs(HttpVerbs.Post | HttpVerbs.Options)]
[AllowCrossSiteJson]
public async Task<ActionResult> Create(MyModel model)
{
    return Json(await DoSomething(model));
}