Try this out and see it works better. The connect method is not documented to accept a DSN in that format.

#!/usr/bin/perl
use strict;
use warnings;
use v5.10;

use DBI;
say "Perl MySQL Connect Demo";

my $db = 'db'; # or whatever it really is
my $server = 'server'; # ditto
my $dsn = "DBI:mysql:database=$db;host=$server";
my $username = "user";
my $password = 'password';

my $dbh = DBI->connect($dsn,$username,$password);
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See also DBD::mysql Class Methods for the full documentation on connect.

Update: You may also need to add in the port.

Update 2: You may need to set some MySQL specific variables in order to get a secure connection working.

...
my $port = 12345; # whatever is should be
my $dsn = "DBI:mysql:database=$db;host=$server;port=$port;
mysql_ssl=1; 
mysql_ssl_client_key=/var/lib/ssl/client-key.pem; 
mysql_ssl_client_cert=/var/lib/ssl/client-cert.pem; 
mysql_ssl_ca_file=/var/lib/ssl/cacert.pem"; # Your paths may vary...
...
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