The Spaniard's 82nd-minute shot was diverted past keeper Craig Gordon by the helpless Phil Bardsley to ensure a topsy-turvy encounter ended all square.
Nineteen-year-old Danny Welbeck looked to have fired the Black Cats into sixth place in the Premier League table with a double either side of half-time as he cancelled out and then overhauled Tim Cahill's early opener.
Welbeck, on loan from Manchester United, might have claimed victory deep into injury time, but he blazed wide from close range, and he should have been made to pay in front of a crowd of 37,331 at the death when substitute Jermaine Beckford passed up a glorious opportunity to win it.
Sunderland have now played every one of last season's top 10 and lost to none of them with only promoted West Brom and Newcastle having collected all three points against them.
In the most unpredictable of starts to the new Premier League season, there was an inevitability about how things would unfold at the Stadium of Light on Monday evening.
The Black Cats returned to home territory still flushed by the euphoria of their stunning 3-0 win at Chelsea last time out and ready to repeat the dose.
Manager Steve Bruce had warned in his programme notes that their heroics in West London would count for little if they did not repeat them, and right on cue, Cahill proved their nemesis, in the short term at least, once again.
The game was only six minutes old when the Australian managed to find enough space inside the Sunderland penalty area to meet Leighton Baines' inviting cross with a firm downward header to make the most of the full-back's neat interchange with Steven Pienaar down the left.
That link-up proved to be the main avenue to goal for the Toffees as Baines and Pienaar caused problems for opposite numbers Nedum Onuoha and Bolo Zenden.
However, the Dutchman turned the tables with 23 minutes gone to pave the way for his side's equaliser.
He skipped away from South Africa international Pienaar on the touchline and deftly manoeuvred his way past Baines to fire a low cross to the near post where Welbeck, who scored his first goal for the club at Stamford Bridge, eagerly slammed the ball home.
Kieran Richardson stabbed a 27th-minute effort just wide with his weaker right foot as Sunderland threatened to make the pressure tell.
But with Everton's left flank continuing to cause problems and Seamus Coleman starting to test Bardsley on the right, the visitors gradually established a foothold once again, and they might have regained the lead 12 minutes before the break.
Cahill timed his run to perfection to meet Baines' corner and head towards goal.
Henderson blocked his effort at the near post, but the ball ran across goal and Lee Cattermole, stationed on the far post, had to scramble it away before the blue shirts could pounce with defender Sylvain Distin eventually firing high over the bar.
The visitors finished the half strongly, but Welbeck headed just over at the death as play switched swiftly from end to end.
Sunderland returned in determined mood and repeatedly put the Everton goal under threat in the opening five minutes of the second half.
Tim Howard had to get down well to keep out Welbeck's curling effort, but he needed the help of first Phil Jagielka and then Arteta to deny Cattermole and Zenden respectively.
But it was Cahill who almost did the damage once again with 55 minutes gone when he was allowed to run on to Phil Neville's ball over the to and lob the advancing Gordon.
The ball looked to be creeping inside the near post until Michael Turner launched himself into a desperate clearance which left him needing treatment.
Everton sensed an opportunity as the home side became increasingly careless in possession, they surged forward in wave after wave.
Bruce introduced Steed Malbranque as a 69th-minute replacement for Zenden, but it was Welbeck who made the breakthrough two minutes later.
Jagielka did well to get a block on Darren Bent's shot to concede a corner and when Richardson crossed at the second time of asking, the teenager looped a header over Howard and into the net.
But with time running down, Everton fought their way back into it when the unfortunate Bardsley deflected Arteta's shot past keeper Gordon.
Welbeck dragged a close-range effort wide deep into injury time, but Beckford perhaps should have won it with virtually the last kick of the game when he ran on to Jagielka's clearance but fired wastefully over with just Gordon to beat.