Two options:
Javascript
Use javascript to construct a url in the form of http://username:password@domain.com upon form submission, set document.location to the constructed url and the browser will redirect whilst automatically authenticating.
Server Side
Allow the form to submit to itself and use server side code to perform the same redirect - I recommend using a 307 redirect.
In php:
redirect("http://username:password@domain.com", 307);
Downsides
Both versions will still result in the web browser showing the ugly authentication box if the initial username and password submission is incorrect.
If you need to avoid this scenario then you will need to implement your own server side authentication scheme.
More detail
If you can tell me which option you believe you will favour, I can show a little more example code.
Security Implications
There are security implications to using HTTP basic auth. If you do not use SSL then passwords will be sent in cleartext. Using the above methods will send the password in cleartext as part of the URL. There are other security implications too:
https://security.stackexchange.com/questions/988/is-basic-auth-secure-if-done-over-https
Because of these I would always recommend against using http basic auth. Many users may be using the same password for sensitive accounts so it's well worth investing a little time in even a basic authentication system.