A victory at the Emirates Stadium would have put United within touching distance of a record 19th league title, but Sir Alex Ferguson's side gave a strangely subdued display and Chelsea can now climb above them into first place if they win next Sunday's crunch clash at Old Trafford.
Chelsea, in second place, trail United by just three points and a win in Manchester would leave them favourites to retain the title with only two games to play after next weekend.
United, who have now failed to win at any of their top-six rivals this season, paid the price for their lethargy when Ramsey struck in the 56th minute.
After starting the weekend firmly in control of the title race, United are in severe danger of surrendering the initiative to a resurgent Chelsea, who have won eight of their last nine league games.
With Cesc Fabregas sidelined by a thigh injury suffered in training yesterday, Arsene Wenger, the Arsenal manager, brought Ramsey into the youngest starting eleven of any Premier League club this season.
Wenger's side had an average age of 23 but that lack of experience did not stop them tearing into United.
They should have had a penalty in the first half when Nemanja Vidic clearly used his hand to deflect Theo Walcott's cross away from Robin van Persie as the Dutchman prepared to connect with a header towards goal, but neither Chris Foy, the referee, or his linesman Andy Garratt spotted the Serbia defender's blatant offence.
United's Wayne Rooney went close with a curling free kick that Wojciech Szczesny kept out at full-stretch early in the second half.
Arsenal took the lead in the 56th minute when Ramsey scored his first Gunners goal since January 2010.
Van Persie twisted and turned in the penalty area before cutting back to Ramsey, who slotted past Edwin van der Sar.
Szczesny had to block Nani's effort and Michael Owen could had a penalty shout turned down when he was tackled by Gael Clichy, but Arsenal deservedly held on for their first victory over United since November 2008, a result which keeps alive their own slender title hopes.