United beat the champions Leicester City 2-1, with goals from Jesse Lingard and new signing Zlatan Ibrahimovic.
Jamie Vardy equalised for Leicester soon after the break, having been a willing beneficiary of a lax pass from Marouane Fellaini - whether the Belgian has a long term future at Old Trafford remains to be seen. Late on, Ibrahimovic netted the victor with a powerful header to open his account on English soil for the club.
Under Louis van Gaal, United were criticised for their sterile domination and lateral passing and, although Mourinho paid a honest tribute to Van Gaal, he was also keen to stress the difference between their styles.
Mourinho has even more reason to be cheerful ahead of the start of the new Premier League season next week.
Speaking to BT Sport after the game on Sunday, Mourinho said: "I think he played very well". He wants to win, and he does everything to win.
"No one remembered they were there because of me but I remember I win this trophy because someone won before me and allowed me to be here".
However, it was Lingard, whose sublime volley settled the FA Cup final against Crystal Palace in May, who opened the scoring when he slalomed past a quartet of attempted challenges before firing a low shot beyond Schmeichel.
Mourinho admitted that it was "difficult" to leave out "good players like Memphis Depay, Phil Jones and Ashley Young", but he says he "had to make choices", before concluding that he told those players that "we are a team and they understand that".
"It feels unbelievable to win the first official game", Ibrahimovic admitted. I'm super happy. This is why I came. "I wouldn't be ambitious if I said it was a fantastic performance". I watched the same team as last season.
However, Ranieri could not fault the work ethic and attitude of his players and believes they are on the right trajectory towards being fully prepared for their Premier League title defence.
"They have signed very well, and if they manage to also sign Pogba, I think they can be real contenders".
The Chelsea boss played down the incident, explaining he needed to replace 28-year-old Mata to "stop the game".