Basically, ajax request as well as synchronous request sends your document cookies automatically. So, you need to set your cookie to document, not to request. However, your request is cross-domain, and things became more complicated. Basing on this answer, additionally to set document cookie, you should allow its sending to cross-domain environment:
type: "GET",
url: "http://example.com",
cache: false,
// NO setCookies option available, set cookie to document
//setCookies: "lkfh89asdhjahska7al446dfg5kgfbfgdhfdbfgcvbcbc dfskljvdfhpl",
crossDomain: true,
dataType: 'json',
xhrFields: {
withCredentials: true
},
success: function (data) {
alert(data);
});