#!perl
# home.pl
use strict;
use CGI;

my $q = new CGI;
print $q->header(),$q->start_html,
$q->start_form(-method=>"post",-action=>"wel.pl"),
$q->b("ID : "),
$q->textfield(-name=>"id", -value=>$q->cookie('id')),
$q->submit(),$q->end_form,$q->end_html; 


#!perl
# wel.pl
use strict;
use CGI;

my $q = new CGI;
my $id = $q->param('id');
my $cookie = $q->cookie(
   -name => 'id',
   -value => $id,
   -expires => '+2m',
   -path => '/');

if ( ! $id ){
  print $q->redirect(-uri=>"home.pl", -cookie=>$cookie);
} else {
  print $q->header(-cookie=>$cookie);
  print $q->start_html,
$q->a({-href=>'home.pl'},'home'),
$q->p("Hello $id"),$q->end_html; 
}
[download]

There are 2 things to be aware of when using print $q->redirect(-uri=>"home.pl", -cookie=>$cookie); as pointed out in the CGI documentation under Generating_a_redirection_header.

1) You should always use full URLs (including the http: or ftp: part) in redirection requests. Relative URLs will not work correctly.
2) All names arguments recognized by header() are also recognized by redirect(). However, most HTTP headers, including those generated by -cookie and -target, are ignored by the browser.

I take from those 2 comments that the results could be unpredictable.

this is the full script

HOME.PL

use DBI;
use CGI;
use strict;


my $cgi = CGI->new(); 
my $db ="databa";
my $usr ="root";
my $pwd ="";
my $host ="localhost";


my $dbh = DBI->connect("DBI:mysql:$db:$host", $usr, $pwd {
   AutoCommit => 0,
    RaiseError => 1,
} )   or die $DBI::errstr;

my $username = $cgi->param('username');
my $password = $cgi->param('password');


my $sth = $dbh->prepare("select id from mysql_auth where username=? AN
+D password=?");
$sth->execute($username,$password)
if(my $x = $sth->fetchrow());
{
$sth->finish();
my $cookie = $cgi->cookie(
   -name => 'login',
   -value => $username,
   -expires => '+2m',
   -path => '/');
print $cgi->redirect(-url=>"welcome.pl", -cookie=>$cookie);
} else {
  print"wrong password\n";
}

WELCOME.PL
use DBI;
use CGI;
use strict;


my $cgi = CGI->new(); 
$sid = $cgi->param('login');
if ( ! $sid ){ # exit and return to home.pl! we have no cookies 
  print $cgi->redirect(-uri=>"home.pl", -cookie=>$cookie);
} else { #start using the page! we have cookies 
  print $cgi->header(-cookie=>$cookie);
 }
[download]