After Sunderland started brightly, the Gunners, who head to Barcelona on Tuesday for the second leg of their Champions League last-16 clash, gained the upper hand, as Nicklas Bendtner's angled shot was tipped over by man of the match Simon Mignolet.
The crossbar saved Sunderland in the second half, while Andrey Arshavin should have been given a penalty and then had a goal incorrectly ruled out for offside.
Arsenal were looking to close the gap on United, who play at Liverpool tomorrow, down to just one point following an unbeaten league run of 10 games.
Sunderland, meanwhile, were out to avoid a fifth successive defeat after showing such promise earlier in the campaign and a first win at Arsenal since 1983.
There was an early chance for Steve Bruce's men when Jordan Henderson got into the right side of the Arsenal penalty area and his cross was cut out by Gael Clichy.
Arsenal could have been forgiven if they had other things on their minds, with the trip to Barcelona on the horizon.
It took a while for the beaten Carling Cup finalists to settle as Sunderland continued to enjoy the better of the early possession.
Arsenal suddenly burst to life on 10 minutes as Bendtner was inches away from connecting with Clichy's low centre across the six-yard box, before Samir Nasri's curling effort was well held by Sunderland keeper Mignolet.
However, it needed a good save from Wojciech Szczesny - whose last-minute blunder cost Arsenal so dear at Wembley - to deny Stephane Sessegnon the opener on 17 minutes after the Benin midfielder was left too much space on the right of the Arsenal penalty box.
The atmosphere inside Emirates Stadium was somewhat muted, as the home crowd grew increasingly frustrated at Arsenal's lack of usual fluid passing game.
Arshavin got ahead of Titus Bramble at the edge of the Sunderland area, only for the ball to run through to the keeper.
Nasri showed great close control to weave into the penalty box, but his near post shot was deflected behind.
From the resulting corner, Bendtner's acrobatic overhead kick went straight at the keeper.
As half-time approached, Jack Wilshere chipped the ball through to Bendtner down the right side of the box, and his half volley was dipping into the far corner - but Mignolet tipped it over.
Arsenal were in need of some inspiration if they were to capitalise on United's midweek defeat at Chelsea.
Sunderland, though, continued to hassle the Gunners at the start of the second half.
Both teams went through a spell of being unable to hold onto the ball, adding to the sense of frustration.
Substitute Marouane Chamakh almost had an instant impact just after the hour as he burst down the right and squared for Arshavin, but his first-time shot was saved at the near post by Mignolet.
Arsenal fans upped the volume, yet Sunderland refused to buckle.
Arsenal had a free-kick just outside the area when Wilshere was tripped by Sessegnon. Nasri lifted it over the wall, but Mignolet touched it onto the crossbar before it was hacked clear.
The same woodwork saved Sunderland again when Chamakh crashed in his header from Wilshere's cross, before Bacary Sagna drilled an angled shot just over.
Arshavin darted onto Nasri's throughball ahead of Bramble, but just as he was about to pull the trigger, the Russian was pushed in the back by the stumbling defender - but the referee failed to give a penalty.
It was a massive let-off for Sunderland, who almost snatched the lead when Danny Welbeck's shot on the turn was pushed wide by Szczesny.
Arshavin looked to have beaten the offside trap as he rounded the keeper, but the assistant on the far side flagged - even though replays showed the Russian level with the Sunderland right-back.
Arsenal continued to press through four minutes of stoppage time, but in the end were left frustrated.