There will be no Champions League football for United, however, for just a second season in the last 25, as they failed to score the 18 goals required to usurp rivals Manchester City into fourth place.
It was a positive way to end of their league campaign, at least, with Rooney in particular proving a real threat by playing a part in all three of his side's goals in this rearranged fixture.
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The closest the home side came to making a breakthrough prior to Rooney's opener came 31 minutes in, when the skipper's free kick landed awkwardly in front of Adam Federici.
Bournemouth's stand-in stopper made a routine enough stop in the end, but Michael Carrick so nearly got a telling touch to the ball following a very bland opening half-hour at the Theatre of Dreams.
United's lowest home league crowd for some time watched on as their side battled to make something happen, although it was proving to be an extremely frustrating evening in the final third of the field.
It could have been even worse for those in attendance had Marc Pugh made the most of his half-chance 38 minutes in, only to send his shot straight at Antonio Valencia in front of goal after some positive play by Matt Ritchie down the right.
The Red Devils did manage to find a way through prior to the break, however, as Anthony Martial linked up with Juan Mata inside the box before picking out Rooney just a few yards out, who had the simple task of turning the ball home for a landmark goal.
Rashford did well to let the ball run through for Rooney's key opener - his 100th at Old Trafford - but that aside it was a half that largely passed the England wildcard by.
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A livelier start to the second period saw both teams come close to finding the game's second goal, with Dan Gosling blasting a yard over the bar and Rooney seeing his latest shot from range this time blocked aside.
Louis van Gaal's men were a little hard done by when Mata, omitted from the Spain squad for Euro 2016 earlier today, went down under a challenge from Steve Cook inside the box but was not awarded a penalty.
Bournemouth were starting to lose a grip on the match as it wore on, and they could well have been two behind when Carrick's bullet shot from 25 yards swerved onto the crossbar and bounced to safety.
As the hour mark approached the hosts squandered two more decent openings - Jesse Lingard sending a shot agonisingly wide of the target just moments after Mata's attempt was blocked aside.
Valencia's shot from all over 30 yards out forced Federici into his latest save of the contest, before Mata smartly twisted and turned and sent an effort down the middle for the keeper to collect.
The one-way traffic showed no real signs of changing and, following some prolonged pressure, United did indeed have a second goal to their name through Rashford.
A sweeping Rooney pass from deep was headed intro a central position by Valencia for the youngster to rifle past Federici, making it eight goals since earning his senior debut just two months ago.
There was still time late on for Young to come off the bench and coolly tuck home a third, latching on to a Rooney chip to seal the points for his side in an impressive manner, although Max Gradel's deflected shot off Chris Smalling at the death did take a little shine off the result.
Bournemouth finish their season in 16th place as a result of this final-game defeat, while United now know that they will contest Europa League football at the very least next term regardless of their FA Cup fortunes on Saturday.