Sources said United will pay Chelsea an initial £35 million, potentially rising to £40m, to reunite Matic with Jose Mourinho, who brought the midfielder to Stamford Bridge in January 2014.
Mourinho told the club's official website: "Nemanja is a Manchester United player and a Jose Mourinho player. He represents everything we want in a footballer; loyalty, consistency, ambition, team player.
"I would like to thank him for his desire to join us because without that, it would be impossible to have him here. I am sure our players and supporters will love him. A big welcome to our new No. 31."
Matic said he intends to repay Mourinho's faith in him, believing the manager will continue to make him better.
"Manchester is one of the biggest clubs in the world (but) also my wish was to train with Jose," the defensive midfielder told MUTV. "Of course when you have a coach like him and a club like Manchester is you don't need to think a lot so my decision was very easy.
"I am very happy because we will work again together. He is a great coach and, when I worked with him before, he always took the best from me and I improve if I work with him.
"When the coach wants you to come to the club, it gives you more motivation, more possibility and I will try to... not try, I will, for sure, give my best to show that I deserve to be in this club and part of this group."
Matic initially joined Chelsea in 2009 but was sold to Benfica in 2011 after making just two Premier League appearances. Mourinho paid £21m to bring him back in 2014, with the midfielder becoming an integral part of Chelsea's title-winning team in 2014-15.
He has already set his sights on replicating the success he had at Stamford Bridge, where he won the league twice -- once with Mourinho.
Matic added: "I think that this squad is ready because it is more mature. They already work with Jose for one year so they know what they have to do and I am sure that this season we can do many things.
"As a member of the team, I think we can do many things. I'm sure that we will try to win the Premier League. Also, the Champions League, you never know what can happen.
"In the Champions League, everything is possible. [There are] many good teams, the best teams in Europe, so Manchester is one of the best for sure and it's very difficult to make a plan before the season starts but we will try to win every game, if it is possible."
The Serbia international, who won a second Premier League title under Antonio Conte last season, leaves Chelsea having played 151 games in his second spell with the club, scoring seven goals.
He moves on an initial fee of £35 million, with the potential for £5m more in add-ons, sources said. He is United's third major signing of the summer after former Benfica teammate Victor Lindelof and Romelu Lukaku.
"With Victor we have been together at Benfica. Lukaku, I didn't play with him because when I left Chelsea [the first time] he just signed there and I didn't have an opportunity to train and to play with him," he said. "But I know him because we have played many times against each other. Both of them are great players and I think they are going to help Manchester to reach the targets."
Matic is neither the first Serb nor the first Nemanja to play for United, following in the footsteps of former captain Nemanja Vidic. A fearsome competitor on the pitch, Matic does not want to disappoint his compatriot when he meets him off it.
"Vidic was a great player and one of the players where I look to be similar because he had a great career," he added. "It's great to be at the same club like him and I will try to do my best to repeat what he does because I don't want that, one day, he tells me, 'what are you doing there?' So I will try to do my best for him to be happy."