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 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's Andy Carroll grabbed the spotlight as they beat Swansea City 2-0.
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.
(Editing by Tony Jimenez)
