Mourinho enjoyed the ideal start to his Premier League tenure at Old Trafford with the Red Devils after their convincing performance against Eddie Howe's side.
Juan Mata opened the scoring just before the break, which was followed by a second from captain Wayne Rooney before the hour mark.
Zlatan Ibrahimovic opened his league account with United's third and final goal, before Adam Smith grabbed a consolation for the Cherries.
After the game, Mourinho shared his delight at his team performance on the coast.
He told MUTV: "The three points, the quality of our attacking football, the quality of our control, the quality of us defensively – we were very compact."
The manager believes that their three points came against "a very difficult opponent" and insisted: "I don’t think there’s any doubt we deserved to win."
Mourinho proclaimed: "[Eric] Bailly and Daley [Blind] played well, the midfield players had an unbelievable complement for each other. With [Marouane] Fellaini I always have the feeling that he was not loved by the general Red people," but said he believes: "If he plays like he’s playing now, they have to love him because he’s being so important for us."
He continued: "I’m really happy. It’s just the start; we need to improve every day. But we won and that’s the most important thing."
A lot has been made of the relationship between Mourinho and Mata, having initially sold the Spaniard to the Red Devils. when at Chelsea a few years ago.
Many expected Mata to be heading for the exit door once again, with his brief appearance in the Community Shield only adding fuel to the fire, however Mourinho has insisted that Mata is part of his plans, and stated how important his opening goal was.
He said: "One goal changes the picture as in the beginning of the second half they tried to put more pressure on us, which opened more spaces."
Mourinho concluded: "The opening goal is always important – it doesn’t mean you win the match but on a psychological level it brings the confidence up and the team could play very well in the second half."
Manchester United don't feature again until later next week when they take on Southampton at Old Trafford on Friday, August 19 with kick-off at 8pm BST.