It is possible to do what you want with mod_perl. For a fuller picture of what you can do, see Writing Apache Modules in Perl and C. (One of the online chapters walks through authentication and authorization.)

However note that your approach should not be used if performance matters to you. Opening a new database connection is a fairly heavy operation, and a standard technique in mod_perl is to cache the connection and not incur this cost. But that will be impossible for you to do because your authentication requires connecting to the database.

Update: I should have mentioned that the default authentication method is Basic, which passes name/password combinations in the clear. By coincidence the newest story on perl.com right now is Integrating mod_perl with Apache 2.1 Authentication which walks through custom authentication schemes in more detail.