## http://mysite/goto.pl ##
use strict;
use LWP::UserAgent;

my $url = 'http://otherwebsite/deeplink.html';
my $ua = LWP::UserAgent->new;
my $response = $ua->post(
	$url,
	[username=>'username', password=>'password'],
);

if ($response->is_success) {
	my $deeplink = $response->content;
	print $deeplink;

} else {
	my $errmsg = $response->status_line;
	....
}