Your usage of the CGI::Application framework is correct. I would however be adding a path to the cookie object - From some of my code:

$self->header_props(
    -cookie     =>  $cgi->cookie(
        -name       =>  'Application',
        -path       =>  '/',
        -value      =>  $session{ '_session_id' }
    ),
    -uri    =>  $uri
);
$self->header_type('redirect');
[download]

Also too, you may want to confirm the return HTTP headers from your application to ensure that indeed the cookie is being set and that the value therein is as intended.