What you want is what the heavy-OO people call a "singleton". You could write one yourself (WARNING! Untested code!):

package CGI::Single;
our @ISA;
@ISA = ('CGI');

my $instance;
sub instance {
  defined $instance or $instance = new CGI;
  $instance;
}
1;
[download]

Now, say "$q = CGI::Single->instance" instead of "$q = new CGI".

Or, CPAN has Class::Singleton, which should let you just say (again, untested!), mixin-style,

package CGI::Single;
our @ISA;
@ISA = qw(CGI Class::Singleton);
1;
[download]

with the same use as before.