I can think of two ways to deal with this.

First, use a SetUID program. You have to be very careful doing that; make sure it can only do the one task it needs to do, and is only executable by the Web server software. To do that, create the Perl script as the user you want it to run as, test it out, and when it works change the group to a group only the Web server is in, and set permissions to u=rxs,g=rx,o= (AKA 4550)

Or, instead of the Web server actually deleting the files, just have it create a flag file somewhere saying they're ready to be deleted, and have a cron job check for flag files periodically, and when it finds them have it delete the files. This is a little bit more complex, but is safer since there's no SetUID program that can be subverted.