Its really hard to search the any combination of keywords in search engine about this because it used by most popular developer wanted a custom autocomplete by ajax.
I've been faced with same issue and searched a bit. I think the solution below is the most convenient way to solve this if you have a login page. If we consider the login submitted via ajax, none of the browsers remember or offer autocomplate feature for user name and password field additionally ask to remember the credentials. But if you use the javascript submit feature (Probably it's not compatible with older versions of browsers), All of the browsers offers to save the username and password except IE. But I've found another javascript tricky for IE to make it offer to save username and password.
In my login page, I've handled the username and password and send them to serverside by ajax request and if the login is succeeded, i submitted the form by the method below otherwise It had been shown an Alert box to the user that the login was failed.
Please Check the link below:
[EDIT]: Link is broken
There is a fixed page about this issue in the page linked, i can not give you another link because of my reputation. Please search for the quotation below in the page:
Look at the fixed page.
Of course,this approach does not fit if you have a login section in the default page because of the form submitting. This causes the page flickering. I wonder if someone has an idea about it?