Dick Advocaat's side held on at the Emirates Stadium to preserve their Premier League status for another season
The Gunners dominated the first half, though they struggled to break the visitors down, with Mesut Ozil and Olivier Giroud's off-target efforts the closest they came to a breakthrough.
Dick Advocaat threw on Steven Fletcher at half-time, and he almost made an immediate impact, only to be denied by David Ospina.
At the other end Costel Pantillimon saved well from Giroud before Fletcher wasted two more great chances to score.
Arsenal had the best of the late chances, but after Sebastian Coates cleared Aaron Ramsey's effort off the line and Billy Jones cleared against his own post, Pantillimon saved from Theo Walcott to preserve a point.
Arsene Wenger's side spent the majority of the opening 45 minutes camped inside the Sunderland half but lacked the composure in the final third to break the deadlock.
The hosts' best chances came as Ozil lashed over the crossbar after getting on the end a lofted pass from Jack Wilshere - making his first start since November due to ankle problems.
Wilshere then set up Giroud with a similar ball, only for the Frenchman to fire narrowly wide of the right-hand post.
Advocaat made a double change at half-time as Fletcher and Jack Rodwell were brought on for Danny Graham and Connor Wickham.
And the changes almost paid instant dividends - Fletcher denied by a low save fromOspina, who then thwarted Patrick van Aanholt at the near post following a superb Jermain Defoe pass.
Sunderland goalkeeper Pantilimon was forced into his first meaningful save of the match in the 56th-minute, diving brilliantly to his right to prevent Giroud from converting Hector Bellerin's right-wing cross.
Still, the visitors continued to cause Arsenal problems going forward and went close to profiting on the counter, however, Fletcher's lobbed effort was turned wide by Ospina.
Fletcher let another chance go begging by prodding a left-wing delivery from Van Aanholt high and off target.
Sunderland's hearts were in their mouths when Ramsey's deflected cross hit Jones in the face and cannoned off the left-hand post.
Pantilimon provided more heroics to deny Walcott but, despite continued late pressure from Arsenal, a gutsy Sunderland hung on to ensure they can breathe easy on the final day of the season, while Hull and Newcastle sweat over their Premier League futures.
Advocaat was visibly emotional as he went to thank the jubilant travelling support after the final whistle.