The Chilean moved to Old Trafford in a swap with Armenian midfielder Henrikh Mkhitaryan heading the other way.
"We've got one of the best attacking players in the world," said Mourinho. "He is very important for us because we want the best possible players.
"Because he has been in England for quite a long time, I think everybody knows the player he is, I think everybody knows what he did at Arsenal.
"I try not to speak about what he did before in Spain and Italy but in the Premier League he has shown the quality he has."
Mkhitaryan made only 10 Premier League starts this season, and it is understood there was a breakdown in trust between him and Mourinho.
Mourinho said: "I think it was a fantastic deal and a fantastic deal is only fantastic if it is good for everybody.
"United and Arsenal made a fantastic deal, I lost a fantastic player, Mr Wenger lost a fantastic player, Alexis changed from a fantastic club to a giant club and Mkhi changed for a fantastic club, so was a great deal for everybody.
"I believe Mkhi is going to be even better than he was with us. With one-and-a half years in England, he has adapted, I think is a good move for everybody, that's why I'm also happy for Mkhi.
"Could he perform better with us? Could I take more from his talent? Maybe. Could he give a little more also to adapt to us? Maybe, but I don't think regrets, he is in our history."