You can run external programs with your choice of system, fork, open, backticks, or exec. For what I think you want, a combination of fork and exec sounds right.

{
    my @prog = ('/path/to/program', @opts, @args);
    my $pid = fork();
    defined $pid or warn "Could not launch @prog" and last;
    last if $pid; # Parent
    # adjust child environment here, if desired
    exec @prog;
}
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fork sets up a child process which then exec's the external program. Similar code can be wrapped up in a sub which acts as your Tk button handler.

Update: You should also make arrangements to prevent the child from going zombie. That can be done by setting up a $SIG{CHLD} handler, or with wait/waitpid. In an event-driven environment, and with the run of the child indeterminate, the SIGCHLD handler 'IGNORE' is probably best if that is usable on your platform.