Firstly, you have to pipe to the script they specify. This can be manually through the config file or in cPanel, go to mail > forwarders. In the destination email box, put this:
Do this for every mail you want to pipe to eSupport. It should look something like this in forwarding overview:
I am assuming both sites are on the same server. Now go to eSupport > Settings > Email. Now you need to add all those emails that you piped and specify the department etc.