use Net::OpenSSH;

my $sshA = Net::OpenSSH->new($serverA,
                             user => $userA, password => $passwordA);

my $proxy_command = $sshA->make_remote_command({tunnel => 1}, $serverB, 22);
my $sshB = Net::OpenSSH->new($serverB,
                             user => $userB, password => $passwordB,
                             proxy_command => $proxy_command);

$sshB->system($cmd1);
$sshB->system($cmd2);
...