Cross domain POST request is not sending cookie Ajax Jquery

You cannot set or read cookies on CORS requests through JavaScript. Although CORS allows cross-origin requests, the cookies are still subject to the browser's same-origin policy, which means only pages from the same origin can read/write the cookie. withCredentials only means that any cookies set by the remote host are sent to that remote host. You will have to set the cookie from the remote server by using the Set-Cookie header.

Please note this doesn't solve the cookie sharing process, as in general this is bad practice.

You need to be using JSONP as your type:

From $.ajax documentation: Cross-domain requests and dataType: "jsonp" requests do not support synchronous operation.

$.ajax(
    { 
      type: "POST",
      url: "http://example.com/api/getlist.json",
      dataType: 'jsonp',
      xhrFields: {
           withCredentials: true
      },
      crossDomain: true,
      beforeSend: function(xhr) {
            xhr.setRequestHeader("Cookie", "session=xxxyyyzzz");
      },
      success: function(){
           alert('success');
      },
      error: function (xhr) {
             alert(xhr.responseText);
      }
    }
);