I might be wrong, but using "exec" directly in CGI script is not very good idea. The "exec" simply replaces current binary with new binary, it does not terminate the process, so the Web server shall wait for this new binary to finish its work, and that might be undesirable.

The appropriate way would be to do fork then exit the parent and do exec in the child. Ideally, the child should also call POSIX::setsid, to detach itself away from Web server, otherwise it could be killed by the server.

Forking also makes possible to pass your password over the pipe. Something like the following could do the trick

my $chld = fork(); # I wan't the CGI to end quickly
die "Can't fork: $!\n" if(!definded $chld);
exit(0) if $chld; # CGI ends here

POSIX::setsid(); # start new session
$chld = open(CHLD, "|-"); # this does another fork internally
die "Can't fork: $!\n" if(!definded $chld);

if($chld == 0)
{
    # this does the same as ">/dev/null 2>&1" in shell
    close(STDOUT);
    open(STDOUT, ">/dev/null");
    close(STDERR);
    open(STDERR, ">&STDOUT");
    # avoid parsing of the arguments by shell, especially if $Trigger_
+Job
    # or $Captured_User_Name contain some dangerous stuff
    exec './processor.pl', '-j', $Trigger_Job, '-u', $Captured_User_Na
+me;
}

select(CHLD); # make CHLD my default output
$| = 1;  # enable autoflushing
print $Captured_Password; # output captured password
close(CHLD); # wait for child to finish it's work
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The code above is not tested. Read more about processes groups and sessions to understand why the switch of session might be good.