If I understand your question correctly, sounds like LWP will solve your problem.

## 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;
    ....
}
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