It was drab opening 43 minutes from both sides with Michael Carrick and Marc Pugh having the only attempts of any recognition until Rooney's historic goal closed the half in style.
United picked up the pace in the second period with Carrick, Jesse Lingard and Antonio Valencia all coming close but failed to add to the scoreline until Rashford's first-time strike and Young's effort lowered the curtain on another Premier League campaign at Old Trafford despite a late own goal from Chris Smalling.
At a snails pace
This match finally managed to get underway 53 hours or so after initial planned after Sunday's incidents and eventual abandonment and it showed with large sections of Old Trafford empty but the ones who did make it wish they hadn't as they were treated to some slow moving football for the majority of the first 45 minutes.
The first real opportunity of any recognition didn't arise until the 32nd minute when Rooney produced an excellent free-kick which looked to find the run of Carrick but he was inches away from getting a touch onto it but it was easy enough for Adam Federici.
Eddie Howe's side then tried their luck when a drilled ball across the box from Adam Francis found Pugh on the edge of the area but his effort was well blocked by Valencia.
The game was in desperate need of some spark and that was managed two minutes from the break when Rooney found the net with historic style.
It was a great team goal as Anthony Martial burst into life playing a great give and go with Juan Mata before playing a ball into the area with Rashford jumping over it which made it perfect for Rooney to fire home his 100th league goal at the theatre of dreams.
Sparked into life
United certainly improved their play as the second half got underway and had three great opportunities to double their lead.
The game against Eddie Howe's side could prove to be the last for Carrick in a United shirt and he nearly finished it in style as his shot from distance had Federici well beaten thanks to a flick from Tommy Elphick but unfortunately Carrick's possible swangsong was denied by the crossbar.
It was then the turn of Lingard as the pinballed around the area eventually falling to Martial who teed it up for the youngster but his effort was just inches shy of Federici's post.
The Australian was called into action moments later when Valencia let rip from some distance and keeper did well to stick up an arm to tip it away from the top corner.
Coming to an end
It had been another season of disappointment for Louis van Gaal's side but they managed to see the season out in style with two more goals.
The first came from Rashford who gave Roy Hodgson a little piece of what he can do as Rooney's ball found Valencia who headed it into the path of the youngster and he made no mistake as he hit it across Federici into the far corner.
Young added a second four minutes from time as Rooney once again showed his quality in his new 'quarterback' role as his ball over the top was perfect for Young who took his time to chest down the effort before hitting it under the arm of Federici with the deflection sending it in.
The contest ended on a sour note as Bournemouth grabbed a consolation right at the death as Lewis Grabban's cut back found fellow substitute Max Gradel whose effort took a wicked deflection off Smalling and past David De Gea to cruelly snatch away the golden glove right at the death.