The Gunners knew that only a win would do if they were to keep the pressure on Liverpool and Manchester City, but it took them until the 72nd minute to find the breakthrough against their already-relegated opponents.
Alexis Sanchez - passed fit to start despite concerns over a thigh injury - scored twice in the space of nine minutes to fire Arsenal to a crucial victory, though, leaving them just one point adrift of the Champions League places going into the final day of the season.
Arsenal went into the match with an imperious record against the Black Cats, and they quickly took control of the contest, with Sanchez blasting an early effort over the crossbar as Sunderland began on the back foot.
The Gunners had the ball in the back of the net as early as the 10th minute when Sanchez's long-range drive struck Olivier Giroud and flew past Jordan Pickford, but the goal was correctly disallowed after referee Roger East ruled that it had come off Giroud's arm.
There was no let-up in the early stages for Sunderland, though, and they needed Pickford to produce a fine save to keep the scores level when he denied Giroud moments later.
Hector Bellerin was the next to come close with a curling effort from outside the area before Granit Xhaka also tried his luck with a couple of long-range efforts as Arsenal were queuing up to get the opening goal of the game.
Sanchez then squandered another half-chance, but Arsenal's firm grip of the game soon began to loosen as Sunderland finally started to settle at the Emirates.
Petr Cech was called into his first save of the evening shortly after the half-hour mark when he kept out a low drive from Didier Ndong at his near post, and the Arsenal keeper was forced into action again shortly afterwards - this time to deny Jermain Defoe from a similar position.
Arsenal did produce another late flurry of chances towards the end of the half, with Bellerin wasting the first of them when he raced through down the right channel only to fire his resulting effort into the side-netting when he had Sanchez in support.
Aaron Ramsey was the next to come close when his snapshot from inside the area was turned past the post by Pickford, and from the resulting corner Rob Holding could only plant a weak header straight at the Sunderland keeper having lost his marker.
Having dominated the majority of the first half Arsenal nearly got off to the worst possible start after the interval when Nacho Monreal's back-pass almost caught Cech out of goal and had the keeper at full stretch to claw it behind.
Sunderland were awarded an indirect free kick inside the Arsenal area as a result, but Sebastian Larsson's inventive flick was nodded over by Xhaka from underneath his own crossbar.
The bizarre start to the half seemed to affect Arsenal as they were slow to get going, but Sunderland failed to make the most of that and Billy Jones squandered their best chance of the second period when he made a mess of his attempted header having been picked out at the back post by Larsson.
Arsenal began to regain control of the game shortly before the hour mark and Ramsey drew another stop from Pickford before Shkodran Mustafi's effort from inside the box was sliced clear by Lee Cattermole right in front of his own keeper.
Welbeck almost made an immediate impact with a low drive that Pickford got down well to save, but there was nothing the Sunderland keeper could do when his goal was finally breached moments later.
Xhaka's pass picked out Mesut Ozil down the left side of the area, and the German played a first-time ball back across goal to leave Sanchez with a tap-in for the opener.
The Chilean could have made it two goals in three minutes when the ball fell kindly to him inside the area, but he attempted to lob Pickford rather than putting his foot through the ball and his audacious effort bounced off the top of the crossbar on its way over.
It was Pickford who was keeping his side hanging on in the match, and he made two more good stops in quick succession as Arsenal kept up their relentless pressure, denying both Xhaka and Iwobi in the space of a minute.
The Gunners did finally get their second goal with just nine minutes remaining, though, and again it was a tap-in for Sanchez as he nodded in from close range after Pickford's save from Giroud saw the ball fall into his path.
With the prospect of goal difference still potentially playing a part in the top-four race, Arsenal refused to rest on their two-goal advantage, and Pickford saved arguably his best stop until last when he tipped Mustafi's header onto the crossbar with only four minutes remaining.
It was the keeper's 11th and final stop of a busy evening, but he still ended up on the losing side as Sunderland were condemned to finish bottom of the Premier League table having now failed to score in 11 of their last 13 outings.
Arsenal, meanwhile, move to within one point of the top four and would require a win over Everton and for Liverpool to drop points at home to Middlesbrough on the final day to secure Champions League football next season.