How to set cookie value with AJAX request?

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);
});