I've read the documentation, and you do appear to be using the function correctly. I agree, it does seem pretty straight forward, and you already made sure that wake-on-lan was enabled.
Try running the program w/ '-d', or do:

Net::Wake::by_udp('XXX.XXX.XXX.XXX', '00:AA:BB:CC:DD:EE') || die ("Err
+or occurred: $!.");
[download]

...to find out if everything is really going as smoothly as it seems.
HTH,
Chris