Thank you.

I needed the 'database=' in the statement. I also had to use the hostname rather than the IP address, however this may be a network issue.

My final connection code looks like the following:
my $dsn = "DBI:mysql:database=$db;host=$host";
my $dbh = DBI->connect( $dsn, $user, $pass, { RaiseError => 1 }) or die ( "Couldn't connect to database: " . DBI->errstr );