Juan Mata scored the first league goal of the new era in the 40th minute following an error by Bournemouth captain Simon Francis and Wayne Rooney headed the second just before the hour.
Zlatan Ibrahimovic, who headed the winning goal in last weekend's Community Shield game against champions Leicester City, then marked his league debut by scoring with a low drive from 25 metres.
United, without world record signing Paul Pogba who was suspended, were denied a clean sheet soon afterwards when defender Adam Smith beat David de Gea with a fierce shot from 12 metres.
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But Ivorian defender Eric Bailly made his Premier League debut and midfielder Henrikh Mkhitaryan was introduced for the last quarter of an hour as the first Armenian to appear in the cosmopolitan league.
Ibrahimovic, in contrast to the 89 million-pound Pogba, cost nothing and was able to maintain his reputation for scoring in a first game in a new league.
"The first objective is to win," he told Sky Sports.
"The second, if I can, is to score. But if I score it's a bonus.
"It was a difficult game. The first half we weren't 100 per cent but second half became easier, there was more space."
Rooney said the combination of four great signings, plus a new manager, would mean "lots of excitement" for United fans, many of whom had been critical of what the team's uninspiring football under Mourinho's predecessor Louis van Gaal.
Dismissed by Chelsea last December, Mourinho took over after the Dutchman was sacked at the end of his second season despite winning the FA Cup in his final game.
"I am very happy," Mourinho said. "Bournemouth always start with great intensity but we kept our control. We weren’t surprised and then we started playing."
Bournemouth manager Eddie Howe felt the defensive mistake for the first goal undermined his side after a promising start.
"I thought it was an even game until the first goal, that was the key moment," he said.
"We dominated the early stages but you can't gift Manchester United goals, which we did."
(Reporting by Steve Tongue; Editing by Brian Homewood and Ed Osmond)
