Chances were at a premium between the two sides, with the best coming in the opening minutes as former Arsenal captain Robin van Persie, now in the United camp, and home striker Olivier Giroud should both have scored.
Arsenal goalkeeper Wojciech Szczesny pushed a bullet header from the Holland forward against the crossbar in the second half of a match which ultimately failed to live up to its billing, leaving Jose Mourinho's Chelsea still top of the league.
David Moyes' visitors were almost gifted a lead inside the first minute when Mikel Arteta gave the ball away to Van Persie, but the one-time Arsenal skipper saw his scuffed shot saved.
French forward Giroud headed an Arsenal corner wide from inside the six-yard box in what was an electric start.
However, the tempo of the match then dropped, with only a deflected effort from Tomas Rosicky causing United goalkeeper David De Gea some concern during the lull.
United - looking to make up ground on the top four following their 2-2 draw at home with Fulham - continued to frustrate Arsenal with a determined midfield display, closing down space quickly when not on the ball.
When Arsenal did get into second gear as the hour passed, Antonio Valencia cleared a header off the line from Arsenal defender Laurent Koscielny.
Giroud just could not get on the end of a great cross from right-back Bacary Sagna before, at the other end, Szczesny tipped a bullet header from Van Persie against the frame of the goal.
De Gea twice saved from Santi Cazorla during the closing stages as Arsenal pressed for a winner, but each team had to be content with a point.
United made two changes following the disappointment of the Fulham result, as Tom Cleverley and Valencia came in, while midfielder Marouane Fellaini was on the bench following a long lay-off with a groin problem.
Arsenal were looking to return to the top of the table following Chelsea's draw at West Brom, and boss Arsene Wenger made two changes in the wake of the 5-1 mauling at Liverpool, with Tomas Rosicky and Kieran Gibbs starting.
Wenger's men had been caught sleeping by Liverpool's fast start on Saturday, and they were almost behind within the opening minute again, as they were at Anfield, when Arteta gave the ball away on the edge of the penalty area.
However, Van Persie, so prolific in the season before he left the Emirates Stadium, scuffed his shot and Szczesny got down to save.
Arsenal then went on the offensive, with a goal-bound shot from Jack Wilshere blocked before Giroud headed wide from five yards following the resulting corner.
A 20-yard strike from Rosicky spiralled up off the back of United captain Nemanja Vidic, who is set to leave at the end of the season, and almost wrong-footed goalkeeper David De Gea.
Van Persie beat the offside trap after 17 minutes down the left channel, but then made a complete hash of his cross, which flew through the penalty area and out of play on the far side.
The tempo of the match dropped before Giroud cracked a 20-yard effort high and wide following a good spell of Arsenal pressure.
There was a brief stoppage when home left-back Gibbs needed treatment following a heavy fall after making a header in the United box - which left Red Devils manager Moyes fuming after play was correctly called back by referee Mark Clattenburg.
United's Brazilian full-back Rafael then also had to be treated after coming off worse in a 50/50 challenge with Giroud.
Cazorla crashed a 20-yard shot into the back of Wilshere as the first half fizzled out.
United were forced into a change at the start of the second half as struggling Rafael was replaced by Rio Ferdinand, who made his first Premier League appearance since November.
Having been frustrated by United's determined midfield display, Arsenal almost snatched the lead after 61 minutes when Valencia's headed clearance under the crossbar thwarted Koscielny.
Szczesny was then quick off his line to head clear as Van Persie closed in on the Arsenal penalty area.
Both sides made a change as the match entered the final 15 minutes, as United replaced Juan Mata with Adnan Januzaj and Arsenal midfielder Alex Oxlade-Chamberlain came on for Rosicky.
Arsenal were close again when Sagna's right-wing cross was just ahead of Giroud as he raced in at the near post.
With 10 minutes left, Van Persie could have won it when he met a floated cross from Rooney at the far post with his powerful header, from which Szczesny produced his heroics.
There was time for De Gea and Cazorla to have their personal battle in the late stages, but neither side had really done enough to earn victory.