Instruct browser to only prompt for saving username/password when they have logged in successfully?
You could write some javascript to check if the credentials are correct, then if they are submit the form.
Double login but then the form would only be submitted if the credentials are correct.
As far as I know, there's no standard way of doing this, although the topic comes up from time to time.
I would imagine you could use JavaScript with PHP to check if the user exists, has the right password entered, then JS would also be used to unhide a checkbox in the form or possibly and simply prompt them to save the login credentials. Let me know if that makes sense.
Now, I see you're asking whether you can disable the inherent browser inquiry in this case. As far as that goes, you could still follow part of my initial idea and then use browser-specific code to disable the question. I'll look into it.