Arsene Wenger's men were unable to capitalize on Chelsea's slip up, as Wojciech Szczesny was twice forced to make important saves to deny former teammate Robin van Persie.
With Manchester City's clash with Sunderland postponed because of bad weather and Chelsea failing to beat West Brom a day earlier, victory would have taken the Gunners to the summit.
However, in a game in which neither side offered enough offensively to win, Arsene Wenger's men were only able to move within a point of leaders Chelsea and two ahead of City.
David Moyes' side, which was frustrated by a late Fulham equalizer Sunday, has now won just one of its last five Premier League outings and remains seventh.
|VIEW FROM OLD TRAFFORD
|By Declan Taylor
Who would have thought, out of Wednesday's games, that this would be the dud? Manchester United, although its performance was abject, will be happiest with this point away from home.
Arsenal attacked for most of the second half but just could not find a way through. It has now taken one point from its last two in the league – and must now welcome Liverpool to the Emirates in the FA Cup on Sunday. It really is crunch time for the Gunners.
In a lively opening, Jack Wilshere was then well tackled by Patrice Evra, having looked to have fashioned a chance on goal, before Olivier Giroud shrugged off his marker and headed wide from a corner.
Van Persie should have tested Szczesny again after 18 minutes, but he sliced an effort way wide of goal after Juan Mata's through ball - much to the delight of the home fans.
Both sides were let down by poor touches in the final third, with Giroud again getting on the end of a corner to head wide after 24 minutes.
With neither side creating many more chances, the only other incident of note before the break was an aerial collision between Rafael da Silva and Giroud.
The Manchester United man landed awkwardly after challenging with the Frenchman for a header and was replaced by Rio Ferdinand at the break.
While the first half ended at fast pace, the second began slowly - with both defenses easily able to cope with what was thrown at them.
The first chance of the half went to Arsenal, with Laurent Koscielny's header cleared off the line by Antonio Valencia from a Santi Cazorla corner a minute after the hour.
Arsenal piled on the pressure from there, with Giroud nudging Bacary Sagna's cross marginally past David de Gea's right-hand post 14 minutes from time before Van Persie's header was tipped onto the crossbar by Szczesny three minutes later.
De Gea was called into action again once more before full time, getting a strong right hand to Cazorla's effort before tipping it around the post.