After thinking about this some more, it is a bad idea to mix the cookie and the form tokens since it defeats the whole purpose of the anti forgery token. It is better to keep the cookie part as a cookie while moving the form part to an auth header, therefore this new answer (again as an AuthorizeAttribute).
using System;
using System.Linq;
using System.Net.Http;
using System.Web;
using System.Web.Helpers;
using System.Web.Http;
using System.Web.Http.Controllers;
  [AttributeUsage(AttributeTargets.Class | AttributeTargets.Method, AllowMultiple = false, Inherited = true)]
  public class ApiValidateAntiForgeryToken : AuthorizeAttribute {
    public const string HeaderName = "X-RequestVerificationToken";
    private static string CookieName => AntiForgeryConfig.CookieName;
    public static string GenerateAntiForgeryTokenForHeader(HttpContext httpContext) {
      if (httpContext == null) {
        throw new ArgumentNullException(nameof(httpContext));
      }
      // check that if the cookie is set to require ssl then we must be using it
      if (AntiForgeryConfig.RequireSsl && !httpContext.Request.IsSecureConnection) {
        throw new InvalidOperationException("Cannot generate an Anti Forgery Token for a non secure context");
      }
      // try to find the old cookie token
      string oldCookieToken = null;
      try {
        var token = httpContext.Request.Cookies[CookieName];
        if (!string.IsNullOrEmpty(token?.Value)) {
          oldCookieToken = token.Value;
        }
      }
      catch {
        // do nothing
      }
      string cookieToken, formToken;
      AntiForgery.GetTokens(oldCookieToken, out cookieToken, out formToken);
      // set the cookie on the response if we got a new one
      if (cookieToken != null) {
        var cookie = new HttpCookie(CookieName, cookieToken) {
          HttpOnly = true,
        };
        // note: don't set it directly since the default value is automatically populated from the <httpCookies> config element
        if (AntiForgeryConfig.RequireSsl) {
          cookie.Secure = AntiForgeryConfig.RequireSsl;
        }
        httpContext.Response.Cookies.Set(cookie);
      }
      return formToken;
    }
    protected override bool IsAuthorized(HttpActionContext actionContext) {
      if (HttpContext.Current == null) {
        // we need a context to be able to use AntiForgery
        return false;
      }
      var headers = actionContext.Request.Headers;
      var cookies = headers.GetCookies();
      // check that if the cookie is set to require ssl then we must honor it
      if (AntiForgeryConfig.RequireSsl && !HttpContext.Current.Request.IsSecureConnection) {
        return false;
      }
      try {
        string cookieToken = cookies.Select(c => c[CookieName]).FirstOrDefault()?.Value?.Trim(); // this throws if the cookie does not exist
        string formToken = headers.GetValues(HeaderName).FirstOrDefault()?.Trim();
        if (string.IsNullOrEmpty(cookieToken) || string.IsNullOrEmpty(formToken)) {
          return false;
        }
        AntiForgery.Validate(cookieToken, formToken);
        return base.IsAuthorized(actionContext);
      }
      catch {
        return false;
      }
    }
  }
Then just decorate your controller or methods with [ApiValidateAntiForgeryToken] 
And add to the razor file this to generate your token for javascript:
<script>
var antiForgeryToken = '@ApiValidateAntiForgeryToken.GenerateAntiForgeryTokenForHeader(HttpContext.Current)';
// your code here that uses such token, basically setting it as a 'X-RequestVerificationToken' header for any AJAX calls
</script>