You could simply define the username/password in your program, and see if it matches the ones the user submits. But this has nothing to do with Perl. You could do this in many other languages.

We are talking about an app with root access but no brute forcing or stuff like that.

Oh, so the password would be the password of a system account? Your question is poorly worded. Why does your application need root access? I would try to avoid that, if I were you. (And don't even think of putting the root password in the program, like I suggest above.) If you must use a system account, then I would create a special account, which only has privileges to accomplish what your application needs done.