Matic joined United on a three-year contract late last month after the Red Devils had reached an agreement with Chelsea.
Mourinho also re-signed the 29-year-old for Chelsea in 2014, and the boss has paid tribute to the Serbian midfielder, who could make his Premier League debut for the 20-time English champions against West Ham United on Sunday afternoon.
"When I joined Manchester United last year, I thought immediately that we needed a kind of Matic. There are not many – there are just a couple – but I thought that," Mourinho told United's official website.
"This season, when I got the call that Matic wants to play for me and for Manchester United, and he can make it happen, I thought he's the perfect player for me. Not just for his qualities but also for my relationship with him, the way I know him, the way he knows me."
Matic has helped Chelsea win the Premier League title in two of the last three seasons.