The Gunners created a host of chances at the Emirates but the closest they came to breaking the deadlock was when Billy Jones deflected a cross on to his own crossbar.
Sunderland also had opportunities of their own but Steven Fletcher twice failed to beat David Ospina.
But the Black Cats will be more than satisfied with a point that guarantees their top-flight status for next season and means either Newcastle or Hull will be relegated on the final day.
The result also all-but guarantees Arsenal third place and automatic Champions League group stage qualification as they are now three points ahead of Manchester United with a superior goal difference.
Sunderland boss Dick Advocaat, who was brought in with nine games remaining to rescue the club from relegation, was left in tears at the final whistle as his mission was completed.
The Dutchman has lost just two matches since taking charge and the draw at the Emirates continued an impressive end-of-season surge that has seen them go five matches unbeaten.
Advocaat made just one change from the draw against Leicester at the weekend as he recalled Adam Johnson to the starting line-up for the first time since February.
Hector Bellerin provided an early threat down the right with a couple of dangerous crosses while Jack Wilshere was just beaten to the ball by a sliding Costel Pantilimon.
The Gunners continued to create chances but none really tested the Sunderland keeper as Mesut Ozil and Wilshere both blasted over the bar and Olivier Giroud also missed the target with a volley.
Sunderland largely wasted the few opportunities they did have on the break and did not have a shot on goal until Johnson shot into the stands late in the half.
Advocaat, who has never been relegated, looked to freshen things up for the second half as Jack Rodwell and Fletcher replaced Connor Wickham and Danny Graham.
The move almost paid immediate dividends as Arsenal goalkeeper Ospina had to be alert to come out and block an angled shot from Fletcher and then stood up at the other post to deny left-back Patrick van Aanholt, who had surged forwards on the overlap.
That scare sparked Arsenal into life, with Giroud getting on the end of a whipped ball in by Bellerin, which Pantilimon turned away at point-blank range, before the Sunderland goalkeeper plucked a goal-bound header from Kieran Gibbs out of the air at the far post.
Arsenal opened Sunderland up again as Ozil picked out Alexis Sanchez in a crowded box, but his shot was blocked.
Sunderland then broke suddenly through Fletcher, but his attempted chip was palmed behind by Ospina.
Fletcher then wasted another excellent chance when Van Aanholt played the ball across the Arsenal penalty area from the left, but could only stab the ball over the crossbar from six yards.
Theo Walcott was the next to go close as he clipped the ball over the on-rushing keeper, but Sebastian Coates came across to hack it off the line.
Sunderland then enjoyed a huge slice of fortune as a cross from the right deflected off Jones and on to the bar.
Arsenal continued to press Sunderland back, with substitute Tomas Rosicky, Giroud and Walcott all going close, but the visitors held on, sparking wild celebrations among the travelling support at the full-time whistle.
Arsenal: Ospina (7), Bellerin (7), Koscielny (6), Mertesacker (6), Gibbs (6), Ozil (6), Ramsey (5), Wilshere (6), Sanchez (5), Giroud (6).
Subs: Rosicky (5), Walcott (6).
Sunderland: Pantilimon (8), Jones (7), O’Shea (8), Coates (7), Van Aanholt (7), Cattermole (7), Larsson (6), Johnson (5), Wickham (5), Graham (5), Defoe (6).
Subs: Rodwell (6), Fletcher (5), Buckley (5).
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