Adam put the hosts in front with a deflected effort, then smashed home a second after Robin van Persie had equalised.
It was the first time United had tasted defeat at the Britannia Stadium, their eighth reverse of the campaign dealing a potentially decisive blow to their hopes of securing a Champions League berth.
In a season of unwanted milestones, with West Bromich Albion, Everton and Newcastle United all ending long runs without victories of some kind against United, this was another - and as if all that wasn't bad enough for David Moyes, Stoke had also come into the match having taken just one point from their previous six games.
The defeat for Moyes' men underlined that for all the feelgood factor generated by Juan Mata's club record £37.1million arrival from Chelsea, it remains clear that United are still some way short of being a top-flight force.
The manager had warned before the game that he would have to carefully handle the fitness of his much-vaunted attacking trio of Mata, van Persie and Rooney. And his injury worries were to grow during the match when Jonny Evans sustained a calf injury and Phil Jones was stretchered off following a clash of heads with Jonathan Walters.
Michael Carrick, who filled in at centre-back, was unlucky to deflect Adam's free-kick past David de Gea as Stoke took the lead in the first half.
After the interval, it took just two minutes for Mata and Van Persie to show what they are capable of, as the Spaniard's expertly-weighted pass found his team-mate, who lifted a first-time effort over Asmir Begovic.
However, Stoke took the lead again shortly afterwards when Adam latched on to a loose ball and sent a 25-yard strike into the corner.
And, despite seven minutes of injury-time, United were unable to find a second response.