The Gunners, looking to get their title challenge back on track after their 5-1 mauling at Liverpool, found the champions in resilient mood as their pressing game stifled the hosts' attacking threat.
United could easily have come away with all three points but for a fantastic reflex save by Wojciech Szczesny who turned former Arsenal captain Robin van Persie's late header onto the bar.
United harried Arsenal high up the pitch from the start - a tactic which nearly paid off in the second minute when Mikel Arteta was caught in possession by van Persie who was one on one with Szczesny but the Dutchman's shot lacked conviction and was easily saved.
Van Persie - predictably booed throughout by the home crowd - had a second opportunity minutes later when he got across Per Mertesacker at the near post to meet Rafael's cross but the German managed to get a vital interception to snuff out the chance.
The United striker spurned a third chance on 18 minutes when he slipped the Arsenal offside trap but opted to cut back onto his less-favoured right foot and skewed his shot horribly wide.
The Gunners - looking to take advantage of Chelsea's slip-up at West Brom - struggled to break down a resilient United defence although Olivier Giroud did squander a free early header, failing to find a telling connection.
The Frenchman was involved in the majority of Arsenal's attacks but did not test David de Gea whose only first-half save of note came on 10 minutes when Tomas Rosicky had a shot which deflected off Chris Smalling and momentarily wrong-footed the United keeper who recovered well and gathered down to his left.
The visitors defended comfortably for much of the first half but were dealt a blow when full-back Rafael - who received extensive treatment after a clash with Giroud - was forced off at the break to be replaced by Rio Ferdinand.
The former England man renewed his partnership with Nemanja Vidic at the heart of the United back four, with Smalling moving to right-back - but the enforced change did little to disrupt the visitors who continued to cope well with an Arsenal side who were unable to run beyond the back line.
United had a great chance to take the lead in the opening minute of the second half when Michael Carrick flicked on Juan Mata's free-kick but Smalling was unable to get a telling touch at the far post.
Arsenal came close to going ahead on 61 minutes when Laurent Koscielny met Santi Cazorla's corner but his header was nodded off the line by Antonio Valencia.
Moments later, Giroud went down in the six-yard box after appearing to be pushed by Vidic but referee Mark Clattenburg waved away the Frenchman's appeals for a penalty.
At the other end Juan Mata was teed up by Cleverley but he dragged his 20-yard effort wide.
Valencia became the first player booked by Clattenburg after upending Jack Wilshere moments before Alex Oxlade-Chamberlain replaced Rosicky. Mata made way at the same time after a largely forgettable 73 minutes, with Adnan Januzaj coming on.
Giroud should have given Arsenal the lead on 76 minutes but he couldn't find the vital touch to Bacary Sagna's cross, Vidic just doing enough to deny the Frenchman.
Van Persie could have punished the miss when he got on the end of a pinpoint centre by Rooney only to be denied by an incredible save from Szczesny, who turned the Dutchman's header onto the bar.
De Gea matched his opposite number seven minutes from time, getting down sharply to his right to deny Cazorla.
Cazorla had another late chance from distance but United's Spanish keeper was equal to it, getting down well to his left as Arsene Wenger's side had to be content with a single point.