Both sides had strong penalty appeals turned down too, first when Arsenal defender Per Mertesacker blocked Jermain Defoe's shot with his arm, then minutes later when DeAndre Yedlin's outstretched arm deflected away Alex Iwobi's shot.
Arsenal finished strongly and Mannone saved from Alexis Sanchez and Danny Welbeck but Sunderland grasped the point that moved them above Norwich City on goal difference with both sides having four matches to play.
Second-from-bottom Newcastle United are a point back with one less game remaining.
"We've got ourselves out of the bottom three now for the first time in many months and the challenge now for the lads is to stay out of it," manager Sam Allardyce said.
"I thought we definitely should have had a penalty. Mertesacker turned his back on the ball with his arms in the air and stopped what looked like a certain goal."
Arsenal stayed in fourth spot, behind Manchester City on goal difference and five points above FA Cup finalists Manchester United, but a 19th consecutive appearance in the Champions League is still not guaranteed as United have a game in hand.
The visitors dominated possession in the first half with Iwobi going close and Mesut Ozil forcing Mannone into fine saves although Van Aanholt came closest with a terrific free kick.
Sunderland were full of running after the break though and had the better chances with Defoe having a powerful effort beaten away by Petr Cech and the former England striker then lobbing just wide.
One brighter note for Arsenal was midfielder Jack Wilshere's first appearance of the season after sustaining a leg fracture that required an operation.
He helped his team finish on the front foot with Danny Welbeck and Sanchez both having efforts saved by Mannone as Sunderland hung on for what could be a vital point
(Reporting by Martyn Herman; Editing by Toby Davis)
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