The Hoops defender set the Clydesdale Bank Premier League champions on their way in the 15th minute with a corner which was forced in by Victor Wanyama, and another set-piece with seven minutes remaining led to Gary Hooper slamming in number two to seal the win.
The Scotland player turned his attention to Thursday's Champions League draw and said: "I would like Manchester United, personally. Whoever it is, it is going to be hard for us and we will need to be prepared. But I would like Manchester United, it would be a Battle of Britain type of game."
He added: "I have watched them but I have never been involved in them so that would be my choice.
"You have to believe in yourself, whoever you get is going to be difficult, we are not making any mistakes about that. We did get a result against Barcelona here but it is different types of football we are up against, it is Manchester United so it would be hard.
"I am not going to say I fancy my chances against Manchester United but I am sure we will compete and do everything we can to get a result."
Manager Neil Lennon, who described Mulgrew's deliveries as "world class" was happy with his side's performance.
He said: "We played well and should have been out of sight before we got the second goal. I was pleased with the way we worked their goalkeeper and our concentration levels.
"There was good attitude, tempo and aggression and we managed to maintain that for the majority of the game.
"We have a four-point cushion with a game in hand, I don't know if we will win that game in hand but we are starting to get some daylight between us and the rest, which is obviously what we want."