Juan Mata, who was substituted after coming on in last week's Community Shield, opened the scoring with a tap-in after Simon Francis' error.
Wayne Rooney headed the second before Zlatan Ibrahimovic marked his United league debut with a goal from 25 yards.
Adam Smith scored a consolation goal for the Cherries.
United were without world-record £89m signing Paul Pogba through suspension, while midfielder Henrikh Mkhitaryan started on the bench, with Mata preferred.
Relive the action from Vitality Stadium
As starts go, this was a highly satisfactory day for United - the Mourinho era began with a solid victory at a ground where they came unstuck under Louis van Gaal last season.
United started slowly and did not register a shot on target until a Rooney effort produced a comfortable save from Artur Boruc in the 27th minute.
Yet it was job done once the visitors opened the scoring with a messy goal and added two more in the space of five second-half minutes to put them in a commanding position.
Mata looked unhappy when he was replaced 30 minutes after coming on at Wembley against Leicester last Sunday but the Spaniard was all smiles as he got Mourinho's reign up and running.
Boruc made a fine block from Mata after Francis' under-hit back-pass but the rebound bounced off Francis into the former Chelsea player's path and he tapped home.
United's second goal was also down to some good fortune - Rooney was well positioned to head home after Anthony Martial's mis-hit shot fell perfectly for his captain.
Ibrahimovic was denied by a one-handed Boruc save but the Swede made it 3-0 with a drilled low finish from outside the area to leave Mourinho smiling.
Ibrahimovic continued his remarkable record of marking debuts with a goal.
The 34-year-old Swede has now scored on his first league starts in England, Italy, Spain and France, and in the Champions League.
Any concerns the frenetic pace in English football's top-flight might not suit Ibrahimovic were dispelled with his well-executed finish to follow a goal at Wembley last week.
It was his third goal in total for his new club after scoring on his first appearance in a United shirt in a friendly against Galatasaray as well as the winner in the Community Shield against Leicester.
Mourinho's oldest signing of the summer could turn out to be his shrewdest.
Bournemouth:
Jordon Ibe - Bournemouth's club-record £15m signing from Liverpool looked a threat coming in off the right and driving through the middle. He also had a goal-bound shot blocked after twisting and turning inside the United area.
Lewis Cook - The 19-year-old midfield signing from Leeds United did not lack confidence and showed enough promise to suggest he could be a success in the Premier League.
Manchester United:
Eric Bailly - Solid league debut and made his presence felt very early on with a no-nonsense challenge on Joshua King. A commanding presence in defence.
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Zlatan Ibrahimovic - Took a while to get going but, when he did, the former Paris St-Germain striker made an impact. Ibrahimovic was not afraid to go in where it hurts and was involved in an early clash of heads with Simon Francis while attempting to connect with Antonio Valencia's cross.
Henrikh Mkhitaryan - Came on for the final 15 minutes with the points in the bag. Replaced Juan Mata, who surprisingly started in place of the former Borussia Dortmund midfielder.
Eddie Howe has been one of the busiest top-flight managers in the transfer market this summer, having brought in six major signings.
And while the Cherries made it hard for United in the first half before Francis' error, they were exposed by ruthless opponents after the break.
There is still work for Howe to do on his team and his next addition looks like being Marc Wilson, the Stoke City defender, for £2m.
Bournemouth conceded 67 goals last season - more than relegated Newcastle - and they needed Boruc at his best to keep United from a more handsome win.
That needs addressing if Bournemouth, who scored when Smith found the top corner from 16 yards, are to stay out of the bottom three this season.
Bournemouth manager Eddie Howe: "It was an even game until the first goal. You can't give Manchester United goals which we did.
"Credit to our boys. We kept fighting away and got the goal back and ended the game very strongly.
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"I thought Jordon [Ibe] did well in flashes and showed what he is going to bring to us. He is a really exciting talent and there is more to come from him."
8: Only Alan Shearer and Frank Lampard (8) have scored more opening day Premier League goals than Wayne Rooney (7).
1: Juan Mata scored his first Premier League goal under the management of Jose Mourinho (no goals in 13 Premier League games for Chelsea in 2013-14, before leaving for Manchester United).
0: Jose Mourinho has never lost an opening day fixture in the Premier League (W7 D1).
11: Bournemouth have now gone 11 Premier League games without keeping a clean sheet.
Bournemouth look to bounce back when they provide the opposition for West Ham's first Premier League game at London Stadium on 21 August (16:00 BST). United welcome Southampton to Old Trafford on Friday.(20:00 BST).
Match ends, Bournemouth 1, Manchester United 3.
Second Half ends, Bournemouth 1, Manchester United 3.
Attempt saved. Max Gradel (Bournemouth) left footed shot from the right side of the box is saved in the centre of the goal.
Attempt blocked. Adam Smith (Bournemouth) right footed shot from outside the box is blocked. Assisted by Benik Afobe.
Corner,  Bournemouth. Conceded by David de Gea.
Attempt saved. Andrew Surman (Bournemouth) left footed shot from the centre of the box is saved in the bottom right corner. Assisted by Joshua King.
Corner,  Bournemouth. Conceded by Marouane Fellaini.
Corner,  Bournemouth. Conceded by Marouane Fellaini.
Attempt blocked. Joshua King (Bournemouth) header from the centre of the box is blocked. Assisted by Charlie Daniels with a cross.
Offside, Manchester United. Ander Herrera tries a through ball, but Memphis Depay is caught offside.
Substitution, Manchester United. Memphis Depay replaces Wayne Rooney.
Offside, Manchester United. Ander Herrera tries a through ball, but Henrikh Mkhitaryan is caught offside.
Foul by Zlatan Ibrahimovic (Manchester United).
Steve Cook (Bournemouth) wins a free kick on the left wing.
Ander Herrera (Manchester United) is shown the yellow card for a bad foul.
Foul by Ander Herrera (Manchester United).
Joshua King (Bournemouth) wins a free kick in the defensive half.
Substitution, Manchester United. Morgan Schneiderlin replaces Anthony Martial.
Foul by Henrikh Mkhitaryan (Manchester United).
Adam Smith (Bournemouth) wins a free kick in the attacking half.
Substitution, Bournemouth. Max Gradel replaces Lewis Cook.
Foul by Wayne Rooney (Manchester United).
Steve Cook (Bournemouth) wins a free kick in the defensive half.
Foul by Henrikh Mkhitaryan (Manchester United).
Charlie Daniels (Bournemouth) wins a free kick in the defensive half.
Foul by Marouane Fellaini (Manchester United).
Adam Smith (Bournemouth) wins a free kick in the defensive half.
Offside, Bournemouth. Lewis Cook tries a through ball, but Benik Afobe is caught offside.
Substitution, Manchester United. Henrikh Mkhitaryan replaces Juan Mata.
Offside, Bournemouth. Charlie Daniels tries a through ball, but Benik Afobe is caught offside.
Juan Mata (Manchester United) wins a free kick in the defensive half.
Foul by Joshua King (Bournemouth).
Goal!  Bournemouth 1, Manchester United 3. Adam Smith (Bournemouth) left footed shot from the centre of the box to the top right corner. Assisted by Lewis Grabban.
Substitution, Bournemouth. Benik Afobe replaces Jordon Ibe.
Substitution, Bournemouth. Lewis Grabban replaces Callum Wilson.
Attempt blocked. Anthony Martial (Manchester United) right footed shot from the left side of the box is blocked. Assisted by Zlatan Ibrahimovic.
Goal!  Bournemouth 0, Manchester United 3. Zlatan Ibrahimovic (Manchester United) right footed shot from outside the box to the bottom left corner. Assisted by Anthony Martial.
Attempt saved. Zlatan Ibrahimovic (Manchester United) right footed shot from outside the box is saved in the top left corner.
Anthony Martial (Manchester United) wins a free kick in the attacking half.
Foul by Andrew Surman (Bournemouth).
Jose Mourinho made a winning Premier League start as Manchester United manager with a comprehensive opening victory at Bournemouth.