How to send a PUT/DELETE request in jQuery?

You could use the ajax method:

$.ajax({
    url: '/script.cgi',
    type: 'DELETE',
    success: function(result) {
        // Do something with the result
    }
});

$.ajax will work.

$.ajax({
   url: 'script.php',
   type: 'PUT',
   success: function(response) {
     //...
   }
});

Seems to be possible with JQuery's ajax function by specifying

type: "put" or type: "delete"

and is not not supported by all browsers, but most of them.

Check out this question for more info on compatibility:

Are the PUT, DELETE, HEAD, etc methods available in most web browsers?

You should be able to use jQuery.ajax :

Load a remote page using an HTTP request.

And you can specify which method should be used, with the type option :

The type of request to make ("POST" or "GET"), default is "GET".
Note: Other HTTP request methods, such as PUT and DELETE, can also be used here, but they are not supported by all browsers.

ajax()

look for param type

Other HTTP request methods, such as PUT and DELETE, can also be used here, but they are not supported by all browsers.

If you need to make a $.post work to a Laravel Route::delete or Route::put just add an argument "_method"="delete" or "_method"="put".

$.post("your/uri/here", {"arg1":"value1",...,"_method":"delete"}, function(data){}); ...

Must works for others Frameworks

Note: Tested with Laravel 5.6 and jQuery 3