I’ve got a situation with a large internal corporate web based application running ASP.NET 3.5 on IIS6 generating 401 “Unauthorised” responses followed by 200 “Ok” responses
I believe you can convince Firefox to automatically send NTLM credentials to a whitelisted set of domains via "about:config" settings - use the "network.automatic-ntlm-auth.trusted-uris" setting. I haven't tried this myself though. I'm not sure there's any equivalent for Internet Explorer.
Unfortunately if you're using something else like Kerberos there does not seem to be a way to avoid the 401.