I have found a dirty workaround for this problem, by inserting an invisible iframe and targeting the form to it:

<iframe src="/blank.html" id="loginTarget" name="loginTarget" style="display:none">
</iframe>

<form id="loginForm" action="/blank.html" method="post" target="loginTarget"></form>

The corresponding JavaScript:

$('#loginForm').submit(function () {
    $.post('/login', $(this).serialize(), function (data) {
        if (data.status == 'SUCCESS') {
            window.location = data.redirectUrl;
        }
    })
})

The trick is, that there are really two requests made. First the form gets submitted to /blank.html, which will be ignored by the server, but this triggers the password save dialog in Chrome. Additionally we make an ajax request and submit the real form to /login. Since the target of the first request is an invisible iframe the page doesn't refresh.

This is of course more useful if you don't want to redirect to another page. If you want to redirect anyway changing the action attribute is a better solution.

Edit:

Here is a simple JSFiddle version of it. Contrary to claims in the comment section, there is no reload of the page needed and it seems to work very reliably. I tested it on Win XP with Chrome and on Linux with Chromium.

Are you able to change the form's action value to data.redirectUrl and let the form submit as usual? This should trigger the browser's prompt to save the username and password.

$(form).submit(function () {
    $.post($(this).attr('action'), $(this).serialize(), function (data) {
        if (data.status == 'SUCCESS') {
            $("form#name").attr('action', data.redirectUrl);
        }
    }
...

Have a read here - why doesn't chrome recognize this login form? .

The important comment is:

Yes, it doesn't work when you remove return false. You will need to rewrite your code. Chrome does not offer to save passwords from forms that are not "submitted" as a security feature. If you want the Save Password feature to work, you're going to have to ditch the whole fancy AJAX login.

So you could maybe consider removing the Ajax and just letting the Form post to login, this will probably be the only way for Users that do not have JavaScript enabled to login with your form too.