In Chrome's settings (navigate to
chrome://settings/ or choose Settings from the... menu?) make sure that you have both "Enable Autofill to fill out web forms in a single click" and "Offer to save passwords I enter on the web" are checked. These are under Passwords and Forms and you'll need to click the link to show advanced settings.
Also check that you are signed into Chrome with the account that you intend on being signed in with and that the sync settings are correct. It could be some weird sync issue where it is saving them until you close and reopen the browser or somesuch.
If all else fails, re-installing Chrome couldn't hurt.