The Gunners had looked set to earn a point at Old Trafford on Sunday, with Henrikh Mkhitaryan scoring against his former club to rule out Paul Pogba's opener for the home side.
An injury-time Marouane Fellaini header snatched a victory for Jose Mourinho's men, however, and continued Arsenal's worst run of away results since a stretch of seven consecutive defeats in 1966.
Arsene Wenger's side went into their meeting with United as the only team in the top four tiers of English football yet to win a single point in an away league game in 2018, having lost on the road to Bournemouth, Swansea, Tottenham, Newcastle and Brighton.
The defeat also means that Arsenal are no longer able to end the Premier League season above arch-rivals Spurs for the second successive season.
Sunday's meeting marked Wenger's final trip to Old Trafford as Arsenal boss, too, a ground on which he has lost more matches at in all competitions than at any other (17).
Wenger will leave the north London club at the end of the 2017-18 season after a 22-year spell in charge of first-team affairs.