Despite dominating, David Moyes' side found themselves trailing to Clint Dempsey's 76th-minute deflected strike.
However, Steven Pienaar headed an 90th-minute equaliser and Nikica Jelavic turned home the winner in added time to extend their unbeaten home league record to 12 matches.
Seamus Coleman fired into the side-netting, Marouane Fellaini headed over a corner and Leon Osman clipped left-footed cross-shot wide from Fellaini lay-off as Everton dominated without threatening.
It took until the 54th minute for Spurs to properly test Everton goalkeeper Tim Howard with Jan Vertonghen's swerving shot tipped over by the United States international.
Osman's side-foot drive from Baines' cut-back sailed just over the crossbar before the England international also drew a low save out of Hugo Lloris.
Having soaked up so much pressure it was almost inevitable the visitors would produce a sucker-punch and it duly arrived in the 76th minute.
Dempsey received the ball with his back to goal 30 yards out, turned and hit a shot which took a slight-enough deflection of Sylvain Distin to send it dipping over Howard.
Substitute Gylfi Sigurdsson hit the crossbar with a curling shot as gaps opened up as Everton pressed for the equaliser they at the very least deserved.
It duly came in the last minute when Coleman crossed and Pienaar headed past Lloris. But there was further late drama as Gibson's pass was touched on by substitute Apostolos Vellios and Jelavic struck to the delight of Moyes, whose exuberant celebrations saw him encroach onto the pitch.