Add the name of the server that hosts the web site to list of computers that the user is allowed to log on to.
If you don't want to do that then you will likely need to modify how the web page (script?) validates the user. The logon event cannot occur on the web server (you should see a logon failure in the security event log), it needs to occur on the domain controller.
Here is a good discussion with programming examples about how to validate the users' credentials against Active Directory.
https://stackoverflow.com/questions/290548/validate-a-username-and-password-against-active-directory