Clint Dempsey's strike with 15 minutes remaining looked to have wrapped up what would have been a hard-earned victory but the home side refused to give in and goals from Steven Pienaar and Nikica Jelavic in the 91st and 92nd minutes respectively completely turned the game around.
Andre Villas-Boas made three changes to the team that started against Panathinaikos on Thursday night.
Hugo Lloris was back in goal, William Gallas returned and captained the team with Jan Vertonghen switching to left-back and Mousa Dembele came in for Tom Carroll as we stuck with a 4-4-2 system.
Everton started well and Sandro was well-placed to block Leon Osman's shot in the second minute. Lloris then did enough at the near post as Jelavic attacked Seamus Coleman's cross.
We enjoyed a decent spell of possession that ended with Dembele's half-volley straight at Tim Howard but from that attack, Everton carved out their best chance on the break, Darron Gibson sent Jelavic clear but Lloris was out like lightning to save at his feet.
Dempsey's free-kick flew straight at US compatriot Howard and Gallas did superbly to flick Coleman's cross away from Jelavic in front of goal as both teams looked for that key opening goal.
Everton stepped up at the midway point. Pienaar, Coleman and Marouane Fellaini all missed the target when well-placed and then Osman's cross-shot drifted wide of the far post.
The home side had penalty appeals for handball waved away against Dempsey and Gallas just before the break.
Everton made a change at half-time with Steven Naismith replacing the lively Kevin Mirallas.
The substitute had a golden chance within two minutes as Leighton Baines struck a free-kick into the wall and then tried his luck again, the second shot running into Naismith's path at the far post, but he sidefooted wide.
Gallas produced another block to deny Jelavic and then Sandro's shot looped just wide off a deflection.
Howard was forced into his first serious save on 56 minutes when Vertonghen hit a piledriver free-kick from 25 yards, the goalkeeper tipping over acrobatically. Vertonghen met the following corner from Walker, heading straight at Howard.
Typically, Everton hit back immediately as Fellaini headed straight at Lloris, Osman angled just over from the edge of the box and the same player then forced a fine save from Lloris, down to his right to tip away a snapshot from 22 yards.
The breakthrough arrived on 76 minutes. Tom Huddlestone, just on for Aaron Lennon, found Dempsey in space 35 yards from goal. Dempsey turned and only had eyes for goal, his shot flicked off Distin and looped over the helpless Howard into the corner.
It was so nearly 2-0 in the 85th minute as Vertonghen got forward again down the left and rolled inside to substitute Gylfi Sigurdsson, he controlled in an instant and aimed for the top corner and watched in frustration as his effort hit the top of the crossbar with Howard beaten.
As the clock ticked into injury time, Everton struck. Coleman crossed from the right and Pienaar timed his run to perfection to head beyond Lloris from 14 yards.
If that wasn't tough enough, the home side found a winner 90 seconds later.
This time substitute Apostolos Vellios flicked on for Jelavic to finish from close range, and there was no coming back from that.
Everton: Howard; Coleman, Distin, Jagielka, Baines; Gibson, Osman (Vellios, 81); Mirallas (Naismith, 46), Fellaini, Pienaar; Jelavic (Heitina, 90+4). Unused subs: Mucha, Oviedo, Hitzlesperger, Barkley.
Spurs (4-4-2): Lloris; Walker, Caulker, Gallas, Vertonghen; Lennonn (Huddlestone, 72), Dembele (Sigurdsson, 77), Sandro, Dempsey; Adebayor, Defoe (Falque, 85). Unused subs: Friedel, Naughton, Livermore, Townsend.
Yellow cards: Everton - Baines, Osman. Spurs - Defoe, Vertonghen.
Goals: Everton - Pienaar (90+1), Jelavic (90+2). Spurs - Dempsey (76).
Referee: Mr K Friend.