The optimism created by Juan Mata's transfer from Chelsea drained away as United were beaten for the third time in their last five league games, losing ground on rivals Everton, who beat Aston Villa 2-1, and Tottenham Hotspur who drew 1-1 at Hull City.
United, who fielded Robin van Persie, Wayne Rooney and Mata in a mouth-watering forward line, were again undone at the back.
Adam rifled home a spectacular winner in the second half, after Van Persie cancelled out his deflected opener, to leave United seventh on 40 points, five adrift of Everton in fifth and four behind Spurs.
Elsewhere, an Adam Johnson-inspired Sunderland recorded a thumping 3-0 win at derby rivals Newcastle United while West Ham United striker Andy Carroll set up both goals but was also sent off in the 2-0 victory over Swansea City.
Southampton dished out more misery for Fulham who dropped to the foot of the table after a 3-0 defeat at Craven Cottage while Cardiff City are second from bottom following a 2-1 win over Norwich City.
United had Rooney back from injury to make his first start since New Year's Day, but their attacking talents came up against one of the game's great levellers.
A howling gale made Stoke a hostile environment for their heralded visitors and after an untidy start, the hosts took the lead with a scruffy goal seven minutes before the break.
Adam blasted a freekick from distance that hit Michael Carrick on the edge of the area and flew into the net with United keeper David De Gea wrong-footed.
Van Persie scored for the second game in succession after 47 minutes to draw them level, steering a tidy finish into the far corner after being picked out by Mata.
Yet the expected comeback victory that United fans became used to seeing under former boss Alex Ferguson failed to materialise and five minutes after the equaliser, Adam thundered a terrific shot into the top corner.
Ferguson's successor David Moyes admitted he is at a loss to explain why they are failing to pick up points.
"I don't know what we have to do to win," he told the BBC.
"I thought we were extremely unlucky. We played well in difficult conditions but we have lost to one wicked deflection and one worldy.
"But we should have been out of sight with the amount of opportunities we had today and we only have ourselves to blame."
RESILIENT OPPONENTS
Everton had to pick themselves off the floor after their 4-0 derby thrashing at city rivals Liverpool, and looked in danger of suffering another morale-sapping defeat when Villa took the lead.
Against the run of play, Leandro Bacuna slotted past Tim Howard after being played in by Christian Benteke.
Despite winning only two of their last six league games, Everton have been resilient opponents throughout the campaign and again dug deep to level when Steven Naismith lashed in after Steven Pienaar's flick-on.
The winner arrived five minutes from time when Kevin Mirallas curled home a free kick from 25 metres.
Tottenham were also fresh from a humiliating thrashing, having lost 5-1 to Manchester City in midweek, but unlike Everton there was no instant pick-me-up.
Hull's seven million pounds ($11.50 million) January signing Shane Long made an instant impact with the opener after 12 minutes but Spurs secured a point when Paulinho swivelled and lashed the ball high into the net in the second half.
"We never took the chances and the positive to me is that we ground it out. If we don't win the game we certainly were not going to lose," Tottenham manager Tim Sherwood said.
Sunderland, who were bottom of the table when manager Gus Poyet replaced Paolo Di Canio in October, moved up to 14th with 24 points after winger Johnson scored his sixth goal in four matches either side of a Fabio Borini penalty and a goal from Jack Colback.
It was a mixed afternoon for West Ham's Carroll who resumed his double act with midfielder Kevin Nolan, setting up two goals for his former Newcastle team mate before the England striker was harshly sent off for allegedly swinging an elbow in the 59th minute.
Carroll has yet to find his shooting boots since returning from injury but his sheer physical menace proved too much of a handful for Swansea who were bullied into submission by the giant England forward.
Carroll's bright start turned sour when he was adjudged by referee Howard Webb to have elbowed Chico Flores in the face after the pair tangled near the halfway line, but the contact was minimal and manager Sam Allardyce said they would appeal.
