Here are a few that have really saved my life:

• CGI - I assume this and DBI need no introduction? :)
• DBI
• Storable - Really handy when you need to serialize data (in a db or otherwise). Take a look at it next time you need to store data in a weird format.
• IO::Socket - Indispensible if you are doing any sort of socket stuff.
• Net::Shared - Shameless plug time :) I really hate IPC::Shareable, so I created this module. Its still under development, but next time you need to use shared memory, give it look; its much more stable than its counterpart.
• LWP - huge time saver.
• File::Find - another huge time saver.
• Apache::Session - a module that makes it much easier to implement sessions. Takes care of most issues for you automatically after its set up.
• Mail::Mailer - My favorite mail sending module.
• HTML::TokeParser - My preferred way to parse HTML.