Fortune, who played for the Red Devils from 1999 to 2006, expects big things of the £40million signing from Chelsea.
Fortune told United's website: "He's a very good player. Someone knows the Premier League and, obviously, the manager knows him so he is someone who can fill in for Michael Carrick.
"He's a fantastic addition to the team and I'm really looking forward to someone who can screen in front of the back four. He plays the role perfectly, like a Claude Makelele role, and, at the same time, can play as well.
"He is a really, really strong player who strengthens the squad and gives us a better chance of winning the league this season."
Matic was delighted to make his bow at Aviva Stadium in Dublin where he sported the number 31 jersey – previously worn by Bastian Schweinsteiger.
"It was nice to play in front of our supporters – I think there was a stadium full of Manchester supporters. That was special for me and I'm happy to make my debut for such a big club like Manchester United.
"I must say thanks to the guys. They’ve helped me to already become part of the team and they are all nice guys. I think the staff have also helped me a lot, and I say thanks to them because they’ve accepted me very well already."
Manager Jose Mourinho, who worked with Matic at the Blues, expects the Serbian to swiftly find his feet at the Europa League champions.
"The way he plays, he uses his brain but of course he needs physicality to be on top of his game. So he needs time but his experience, intelligence and genius in the way he thinks about footbal will help him to be ready sooner rather than later,” Mourinho told MUTV.
United’s next challenge will be the highly anticipated UEFA Super Cup match against Champions League winners Real Madrid on August 8.