Submitting passwords via ajax

given the following scenario: We have a html form for changing an account's password. It looks like this:

CurrentPassword:   __________________
NewPassword:       __________________
NewPasswordAgain:  __________________

We want to send this request via an ajax call. If we send it and we leave our computer (without logging out and staying on the exact same page) someone could open the webkit inspector (or firebug) and see something like this:

http://cl.ly/3y213W1q0U2y2e251k0O

What would be your solution for making this more secure? Is it even possible using an ajax call here or would it be better to use a "normal" html form which reloads the whole page after sending?