I'd use Mail::Mailer;

sendmail ('god@heaven.com', 'die.puny@earthling.net','Test', 'This is 
+a test');

sub sendmail {
    my ($from, $to, $subject, $body) = @_;
    my $mailer;
    eval { 
        $mailer = Mail::Mailer->new('sendmail');
    };
    if ($@) {
        warn "Couldn't send mail: $@";
    } else {
        $mailer->open({ From => $from,
                        To => $to,
                        Subject => $subject });
        print $mailer $body;
        unless (close ($mailer)) {
            warn "Couldn't close mailer: $!";
        }
    }
}
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Also, check that whichever user is sending the mail is allowed to set the 'From:' field.
For instance, if you're using exim, add them to the 'trusted users' section of /etc/exim.conf