The result ensures the Premier League champions will take the defence of their trophy into the final day.
The result keeps United level on points with Manchester City, but United have an eight worse goal difference ahead of next weekend's final weekend. United, who travel to Sunderland, will be hoping to better City's result at home to QPR or else they will need a win with an eight-goal swing to retain their crown.
Asked on TV prior to kick-off whether City now had one hand on the trophy after their 2-0 win at Newcastle, Sir Alex Ferguson ruefully replied "two, probably".
There was no doubt the events in Newcastle had an effect on the crowd, who were subdued in the extreme and failed to respond when goaded about their plight by gleeful Swansea counterparts, making their first Old Trafford visit since 1983 and already preparing for another trip next season.
It took United's players some time to shrug off their pain too and Nathan Dyer wasted a decent chance to put the visitors ahead.
However, once the Red Devils got going, there was a purpose to their play.
Had they enjoyed a bit of good fortune in front of goal, they might have made inroads into the 10-goal goal difference advantage City enjoyed at kick-off.
Wayne Rooney had a volley turned away by Michel Vorm, Javier Hernandez wasted two close-range opportunities and Chris Smalling sent an instinctive header wide as the hosts set up camp around the Swansea penalty area.
As it was though, by half-time, all they had managed to do was cut out the goals Yaya Toure scored at the Sports Direct Arena.
Scholes claimed the opener, flicking home a Michael Carrick shot that was bound for the opposite corner.
It was the 37-year-old's fourth Premier League goal since he came out of retirement in January, his highest haul for five years.
Young got the second, latching onto the loose ball after Rooney's shot had been blocked.
With Antonio Valencia causing problems on the other wing, it merely raised obvious questions over Ferguson's team selection at the Etihad Stadium on Monday, when defeat handed over the leadership and - in all probability - the championship.
Not that Ferguson will ever be deflected from the view that United tossed away their chances in that barmy couple of minutes against Everton on their last appearance at Old Trafford a fortnight ago, when a 4-2 lead became a 4-4 draw.
That Young chased into the goal to grab the ball after his goal suggested there was some purpose about United and they were quickly into their stride after the break.
Rooney's attempt to roll a pass across goal to give Young a tap-in was intercepted by Dyer.
Hernandez was inches away from heading home before Valencia smashed a shot goalwards, forcing Vorm into another save.
Yet the loose defending that has been evident far too often this season was also apparent.
Allen was given far too much room to shoot on the edge of the area and when his shot was deflected by Danny Graham, it took an impressive manoeuvre from David de Gea to keep it out.
Dyer then curled a shot over when he should have at least kept his effort on target, and Swansea came close again when De Gea turned away Gylfi Sigurdsson's free-kick.
The introduction of Tom Cleverley for Scholes midway through the second period should have provided United with some additional energy, although it took a while for it to become evident.
In that time, an agitated Rooney picked up his first Premier League booking for a foul on Luke Moore.
It was another of those days when United's star striker was not in the best of moods, and chipping wide after being sent clear by Carrick did not improve matters.
Ferguson had already decided Rooney's performance was going to be cut short anyway, for what is likely to be Dimitar Berbatov's last appearance in a United shirt.
At £30.75million, the Bulgarian represents United's last big-money purchase.
But if, as even their staunchest fans now think, the championship heads across town next weekend, they will expect the Glazer family to loosen the purse-strings once more this summer.
And a generally disappointing day ended on a low note for Rio Ferdinand, who came off near the end after suffering what appeared to be a hip injury that could have implications for his hopes of featuring at Euro 2012.