As noted above, my $string = "üöä"; makes the string contain literally ampersand-u-u-m-l-semicolon etc. Your script says use utf8; already, so you can use literal unicode characters in it, or chr(XXXX) if you want to be extra safe. If you just used character entities here for fear that perlmonks will corrupt your accented characters, well, obviously, the character entities don't get translated if they are in <code> tags.
in reply to printing unicode to STDOUT
On another note, I can't see you OS mentioned here. If you're on *nix, your characters will probably show up correctly in the console without fooling with the encodings at all. If you're on Windows, you might not be able to get them to show up correctly, whatever you do. Maybe you can get it to work on your own computer, but if you want it to work on any windows box of unknown localization, you're in for an uphill struggle.