After a scrappy opening period, Adam gave the home side the lead somewhat against the run of play, and in fortunate fashion, with his long-range free-kick deflecting past David De Gea in the United goal.
But straight after the interval Robin van Persie levelled proceedings, and gave United the initiative.
However, just five minutes later, Adam rifled an unstoppable effort from 25 yards which curled away from De Gea into the top corner, and stunned the champions.
An incredible fingertip save from Asmir Begovic kept out a late Wayne Rooney free-kick as the champions pressed for an equaliser, but the Potters held on for their first league win against United since Boxing Day 1984, and condemned David Moyes' side to their eighth league defeat of the season.
The wind was playing havoc in the early stages, with neither side able to get the ball down and create anything in the final third.
Patrice Evra's drilled cross was fired just wide by Rooney on the half-volley as United started to exert some influence on the game.
The visitors suffered an early blow as Jonny Evans had to be withdrawn with a knock, and endured further woe as they fell behind to Adam's fortunate free-kick.
From fully 30 yards, Adam fired a powerful low effort towards goal, which Michael Carrick could only get a knee to, diverting the ball past a stranded De Gea, and giving the home side a surprise lead.
The goal stunned Moyes' men into action, but for all their pressure and possession they could not find a way through a solid Stoke rearguard.
Moyes' plan was dealt a further blow as Phil Jones had to leave the field on a stretcher after a nasty fall, with Danny Welbeck coming on, Carrick dropping into central defence, and Rooney reverting to a midfield role.
However, just minutes after the restart, a long ball was not dealt with by Wilson, clearing only as far as Juan Mata, who steered the loose ball through to Van Persie, who calmly slotted past Begovic to improve the United faithful's mood.
The goal seemed to give United the impetus, but out of nowhere, Adam fired a powerful strike from distance past De Gea to immediately regain the lead for the home side.
United pressed and pressed for an equaliser, with a wealth of attacking options at their disposal, but they could not find a way through.
Time and again both full-backs managed to get to the byline, pull the ball back, but some last-ditch defending meant the United strikeforce could not convert.
De Gea had to tip substitute Oussama Assaidi's volley over the bar to keep United in with a chance of getting something out of the game.
Stoke needed their goalkeeper to save them at the death, Begovic acrobatically clawing Rooney's expertly-taken free-kick onto the post, with Tom Cleverley's skied loose ball summing up United's day.
Gone are the days when United salvaged points late in the day, having not managed to retrieve any points at all this season in the final ten minutes of a match.