The above is pretty accurate.

What I use is $db = "dbi:mysql:dbname:hostname:port"; so then:
my $dbh = DBI->connect($db, $user, $auth);

perldoc the DBI.pm, because I think you can do the hostname settings inside the connect call as well, as in
my $dbh = DBI->connect("dbi:mysql:dbname", $user, $auth, $hostname, $port);
but my memory is hazy on that one.

Best bet - pod2text DBI.pm and read! OR buy Randy J. Yarger's wonderful book, mSQL and MySQL, from O'Reilly. It covers this question in full, with plenty of examples.