It feels strange to answer to myself. But here is an answer which describes the best my original problem.

It is not possible to transfer cookies between 'agents'. That means if you get cookie via curl, you cannot supply it back to JIRA/REST via LWP.

Use cURL to retrieve original cookie? Then continue with cURL. If you plan to use LWP or JIRA::Client::Automated, like me, get cookie via this agent. See following code:

use LWP::UserAgent;
use HTTP::Request;
my $user = "JIRAusername";
my $password = "Lamepassword";
my $jira_url = "put_your_server_here";
my $ck=LWP::UserAgent->new( );
my $getline= $jira_url . '/jiraurl/rest/auth/1/session'; 
my $request = new HTTP::Request (GET => $getline, HTTP::Headers->new('
+Content_Type' => 'application/json') );
$request->authorization_basic($user, $password);
my $answer = $ck->request ($request);
my $cookie = $answer->header('Set-Cookie');
[download]

The very last line gave me a lot of grey hair, so enjoy it, please. Now you have all cookies in variable. JIRA gives me normal 'JSESSION cookie' and token together. Authentication was NOT functional in case I have used only one of them.

Second pitfall - where and how to put cookie? I mean as a part of JIRA request.

Following piece of code is showing easy way to use it. I have created another lwp agent, just to show cookie is functional for independent one.

my $ua=LWP::UserAgent->new( keep_alive => 1 );
my $key = 'existing_issue_code'; #this is just for test. You can use a
+ny issue you get from web JIRA
$getline=$url . '/rest/api/latest/issue/' . $key;
$request = new HTTP::Request (GET => $getline, HTTP::Headers->new('Coo
+kie'=> $cookie ) );
$answer = $ua->request ($request);
[download]

Then just test return value, it shall be 200 and enjoy...

I will spend some time by debugging of updated version of JIRA::Client::Automated and post it on CPAN later.