Adam put the hosts in front with a deflected effort, then smashed home an excellent second after Robin van Persie had levelled straight after the break.
It was the first time United had tasted defeat at the Britannia Stadium too, their eighth reverse of the campaign dealing a savage blow to their hopes of securing a Champions League berth for next season.
In a season of unwanted milestones, with West Brom, Everton and Newcastle all ending long runs without victories of some kind against United, this was another.
Given Stoke had taken just one point from their previous six games, it represented yet more pain for manager David Moyes, whose team could find themselves nine points off a Champions League place by the end of the weekend.
It also underlined that for all the feelgood atmosphere generated by Juan Mata's club record £37.1million arrival from Chelsea, it is going to take far more to turn United back into a top-flight force.
Although Moyes stated in the build-up that the fitness of his much-heralded attacking trio needed to be managed, he still decided to pick them for the most physical of tests.
Their task was not made any easier by a fierce wind, strong enough to hold the ball up when it was in the air.
If Moyes' idea had been to take one of the key men off before the end, it was scuppered by a couple of injuries in a first half that merely reinforced the feeling his time at United is being dogged by bad luck.
Jonny Evans was first to succumb, forced off with what appeared to be a calf injury before the contest had even begun to take shape.
The visitors did have cover for that eventuality.
But when Phil Jones had to be stretchered off following a nasty clash of heads with Jonathan Walters, with Nemanja Vidic suspended and Rio Ferdinand not making the squad, there was none.
It meant Michael Carrick was forced into central defence alongside Chris Smalling in just the game you would not choose to be weakened.
In between the substitutions came another piece of ill fortune for the visitors.
It was debatable whether Adam's long-range free-kick was even on target when it struck Carrick on the edge of the box.
Once it had, David de Gea had no chance of keeping it out and the ball rocketed into the far corner, leaving the United keeper stranded.
At times like that, Moyes must feel cursed.
Yet, it had to be said, his team were not exactly dominating their opponents, with Mata a largely peripheral figure and neither Wayne Rooney nor Van Persie threatening with any regularity.
Van Persie came close once, with a close-range header that rebounded over off Glenn Whelan.
Rooney had a couple of long-range efforts but with his team struggling to cope with the conditions, it was a surprise Welbeck was introduced for Jones given Antonio Valencia would have allowed Mata to move into a central position, or Darren Fletcher would have kept Rooney in an attacking one.
It took just two second-half minutes for Mata and Van Persie to show what they are capable of.
Marc Wilson's poor clearance from inside his own box offered Mata the chance to turn a perfectly-weighted pass to Van Persie, who lifted a first-time effort over Asmir Begovic and into the corner.
It was the Dutchman's sixth goal in seven Premier League games, split by that thigh injury - and a period in which United's limitations were laid bare.
And so they were again shortly afterwards. For barely had United time to reflect on parity than they were behind again, with Adam once again the scorer.
This time there was no fortune either as he let fly following a blocked Marko Arnautovic shot, sending his 25-yard strike arrowing into the corner.
Arnautovic came close to adding another just after the hour mark as the contest heated up and Rooney's fuse shortened in response to a couple of debatable decisions from referee Neil Swarbrick.
United did their best to respond. But - after Ashley Young had wasted an excellent chance at the far post and Rooney's free-kick been pushed onto a post by Asmir Begovic - there was no redemption, not even in seven minutes of injury-time as Tom Cleverley wasted the final opportunity.