The Net::Telnet documentation, especially its synopsis shows the following code:

use Net::Telnet ();

$t = new Net::Telnet (Timeout => 10,
                          Prompt => '/bash\$ $/');
$t->open("sparky");
$t->login($username, $passwd);
@lines = $t->cmd("who");
print @lines;
[download]

I interpret this code as

1. logging on to a machine "sparky", with a username of $username and a corresponding password
2. executing a command, who, and capturing its output in @output
3. printing the captured output

I'm not sure where the synopsis of Net::Telnet diverges from your stated requirements, or where you've found flaws in the operation of Net::Telnet for your required operation, so I can't help you much further. If you have problems with writing the lines captured in @output to a file, consider looking at open and print, but neither of these have anything to do with a telnet connection.