Chances were few and far between, 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 also pushed a bullet header from the Netherlands forward against the bar 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.
"I tried to react to Robin van Persie's effort," said Szczesny. "He did not score a lot of headers at Emirates Stadium, so I wanted that to continue. We are confident going into the last few games. If we show our character, we can do it."
David Moyes' visitors were almost gifted a lead inside the first minute when Mikel Arteta gave the ball away to Van Persie, but the Dutchman saw his scuffed shot saved. Giroud then headed an Arsenal corner wide from inside the six-yard box in what was an electric start.
However, the tempo of the match soon dropped, with only a deflected effort from Tomas Rosicky causing United goalkeeper David De Gea some concern during the lull.
"We can take a lot from the game," said Michael Carrick, the Manchester United midfielder. "Our shape was good, David de Gea had to make some saves but we had some good chances ourselves. It is not a bad draw, but we came here to win. You have to respect Arsenal, as a point is not too bad."
United had made two changes following the disappointment of the weekend's draw with Fulham, as Tom Cleverley and Antonio Valencia came in, while midfielder Marouane Fellaini was back on the bench following a long lay-off.
Arsenal were looking to return to the top of the table following Chelsea's draw at West Brom, and manager Arsene Wenger made two changes in the wake of the 5-1 humiliation 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.
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. Santi Cazorla also 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 the struggling Rafael was replaced by Rio Ferdinand, who made his first 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 Laurent Koscielny.
Szczesny was then quick off his line to head clear as Van Persie closed in on the Arsenal penalty area.
With 10 minutes left, Van Persie might have won it when he met a floated cross from Rooney at the far post with his powerful header, from which Szczesny again produced his heroics.
"I am pleased, but I wanted all three points," said Moyes. "Arsenal are having a great season, so it is not a bad draw for us. The way my luck was going I expected Mesut Ozil to score with a free-kick at the end.
"I am quite pleased with a lot of things. I do not like talking about luck. But you hope it turns around."