Robin Roefs was Sunderland's hero as Everton's FA Cup hopes came to a premature end once again after three woeful penalties in a shoot-out saw the Black Cats reach round four.
David Moyes' threadbare squad looked like exiting in normal time until James Garner's 89th-minute spot-kick cancelled out Enzo Le Fee's brilliant first-half strike.
But extra time was the last thing the hosts, down to just 12 senior players, needed and when the shoot-out came around Garner, Thierno Barry and Beto all saw their efforts saved by Roefs.
Le Fee, Granit Xhaka and Luke O'Nien confidently scored their first three attempts to help Sunderland into the fourth round for only the second time in 11 seasons.
It was a hammer blow to Everton, who were dragged back into the game by Adam Aznou, a £7.8m signing from Valladolid in the summer who had not had a sniff of first-team action.
The 19-year-old's only senior football was two appearances in the EFL Trophy back in September and October and his 86th-minute introduction for Merlin Rohl was only because Moyes had little to no option.
But the Spanish-born Moroccan showed more drive in the four minutes of regulation time than many of his battle-weary team-mates had in the preceding 86 - particularly fellow 19-year-old and summer arrival Tyler Dibling, who cost five times as much.
Within three minutes he won an admittedly soft penalty after a shoulder-to-shoulder challenge by Trai Hume allowing Garner to equalise.
Replays of the foul on the big screens at full-time enraged Xhaka, who hurled his water bottle into the turf and raged at the perceived injustice to anyone in earshot, but the Black Cats' captain was soon celebrating.
It is rare for Sunderland, promoted from the Championship last season, to be favourites in any game but the fact Regis Le Bris was able to make five changes and still have the likes of Dan Ballard, Brian Brobbey, Omar Alderete and Wilson Isidor in reserve swung the odds in their favour.
Injuries, the Africa Cup of Nations and suspensions for Michael Keane and Jack Grealish were compounded by Tim Iroegbunam's hamstring problem which took the absences to nine.
Barry, the one senior player he left out, accounted for 23 of the 27 first-team appearances among the substitutes with five of six academy players not having previously featured at all.
After Harrison Armstrong sliced wide their only first chance of the first half, Sunderland took control and should have been comfortably ahead by the interval.
Their first shot resulted in the opener - and what a goal.
Everton failed to deal with a long throw and Eliezer Mayenda teed up Le Fee to send a cushioned right-footed volley curling past Jordan Pickford.
Moyes had a face like thunder and his demeanour was not improved as Pickford clawed away a Nordi Mukiele overhead kick and tipped away, at full-stretch, Romaine Mundle's strike from distance.
Surprisingly, the Everton boss did not make the only change available to him and sacrifice one of his five midfielders for Barry at the start of the second half, but they did not come to life until Mayenda rolled an angled shot just wide with less than half an hour remaining.
Beto's appeal for a penalty for a shirt pull by Hume was waved away but referee John Brooks obliged when Aznou went down and Garner equalised.
Unfortunately for Everton their subsequent penalties were not as confident, with Barry and Beto's stuttering run-ups far from convincing, as Roefs saved all three.
Everton boss David Moyes:
"I don't think you blame anybody when you lose a penalty shoot-out. It's something that can happen, the goalkeeper is there to make saves.
"But I think there is a way to lose if you're going to do that. It felt as if we didn't do it in that way and that's why we have all walked away with huge disappointment.
"We did a great job to get back in the game and get ourselves level with what was available but when we get to that point you're hoping to go on and show it and it was poor.
"Someone has to win a penalty shoot-out and someone has to lose it but I do think there is a way to win it and a way to lose it.
"I didn't think the way we lost it today was the right way."
Sunderland boss Regis Le Bris on goalkeeper Robin Roefs:
"He is getting better. He is really young and he wants to learn so I think when you give him opportunities to play at that level he will grow quickly and I think it is only the beginning.
"He will have ups and downs, which is normal, but he is really consistent and helped the team for the penalty shoot-out.
"Robin made three impressive saves but we had to score three goals as well."
