A first-half strike from Juan Mata and goals after the interval from Wayne Rooney and Zlatan Ibrahimovic condemned the Cherries to defeat, despite a great consolation from Adam Smith reducing the arrears.
The Cherries were far from overawed by United's starpower, with the attacking talent of Ibrahimovic, Rooney and Anthony Martial well shackled by Eddie Howe's men for the majority of the first-half.
It took 25 minutes to see the game’s first shot on target and it was the attacking riches of United that combined, as Mata found Rooney who curled a shot inside the area but Artur Boruc was able to hold the effort.
But after more than matching the Red Devils for most of the first-half, the Cherries were made to pay for a defensive lapse six minutes before the break.
Uncharacteristically, skipper Simon Francis under hit a simple back pass which allowed Mata in on goal. His stabbed effort was very well saved by Boruc but the ricochet bounced off Francis' shins which allowed the Spaniard to tap the ball in.
It was a cruel turning point in the game as Jose Mourinho’s side looked more composed from that point onwards.
The Cherries, for all their good efforts, were almost undone again after 51 minutes, when Rooney burst past Cook and forced Boruc into an excellent save from ten yards.
The visitors also had a penalty appeal turned away when Rooney hit the deck in the box before the Cherries created an excellent opening of their own.
Debutant Jordon Ibe picked the ball in the box, beat two challenges before smashing an effort which was extremely well blocked. That lifted the home fans to their feet but it was the away supporters celebrating just before the hour mark.
Antonio Valencia and goalscorer Mata combined on the right hand side and the Ecuadorian put in a ball, which Martial volleyed into the ground at the back post and Rooney was alive to nod in a simple header from six yards.
Four minutes later, Ibrahimovic showed his class which resulted in the third goal – he was yet to take a touch in the penalty area but from 25-yards, he lined up a powerful effort which flew past Boruc into the bottom left corner.
Howe then made his first substitutions of the afternoon, bringing on Benik Afobe and Lewis Grabban for Callum Wilson and Ibe.
Grabban made an instant impact and the Cherries had a lifeline after 68 minutes, his clever reverse ball found Smith in the box and the full back smashed a wonderful effort from 14-yards which gave David De Gea no chance to reduce the deficit.
Despite Smith’s strike, the Cherries couldn’t create any late drama as they began the season with defeat.
AFC Bournemouth: Boruc, A Smith, Francis, S Cook, Daniels, Arter, Surman, L Cook (Gradel 81), Ibe (Afobe 67), King, Wilson (Grabban 67).
Substitutes: Federici, Ake, B Smith, Gosling.
Manchester United: De Gea, Valencia, Bailly, Blind, Shaw, Herrera, Fellaini, Mata (Mkhitaryan 75), Martial (Schneiderlin 84), Rooney, Ibrahimovic.
Substitutes: Romero, Rojo, Carrick, Memphis, Rashford.
Attendance: 11355 (1392 away)
Referee: Andre Marriner
Booked: Herrera
