$ perldoc lwpcook
[...]
COOKIES
       Some sites like to play games with cookies.  By default
       LWP ignores cookies provided by the servers it visits.
       LWP will collect cookies and respond to cookie requests if
       you set up a cookie jar.

         use LWP::UserAgent;
         use HTTP::Cookies;

         $ua = LWP::UserAgent->new;
         $ua->cookie_jar(HTTP::Cookies->new(file => "lwpcookies.txt",
                                            autosave => 1));

         # and then send requests just as you used to do
         $res = $ua->request(HTTP::Request->new(GET => "http://www.yah
+oo.no"));
         print $res->status_line, "\n";
[...]
[download]

Looks rather complete and simple to me. HTTP::Cookies is also well documented.