Over the years, Perl has shown itself to be a very versatile tool that's really good at doing a lot of the things that we need to do all the time. Ripping strings (and HTML data-streams apart), doing efficient data manipulation with hashes and lists, object-oriented programming, and so on. Its architecture has allowed for the vast CPAN library to be built ... and to contain thousands of entries covering everything imaginable. You can exploit that existing code-base in anything.
Various other strong contenders of course exist. You can build a great web-app in almost anything. But Perl has been referred-to as “The Swiss Army® Knife of computing” for a very good reason. It's the tool that you find yourself picking-up again and again and again.
“I need to open this envelope...” okay, Perl can do that. “Now I need something that slices, and dices, and makes Julienne fries...” okay, Perl can do that too.
So, really, when you're looking for “a Perl success story,” I think that you really need to expand the scope of what you are talking about. It is very advantageous when one tool can be applied to so many aspects of what a data-processing shop is actually doing, all within the auspices of the same environment, all with the same underlying code-base, all at the same time. It is “a huge win” when you can have... One Ring to Rule Them All. Where the front-end web applications, and the back-office interactive applications, and the batch jobs, all leverage the same code-base all at the same time.