Everton scored twice in injury time to come from behind and beat Tottenham Hotspur 2-1 in the Premier League at Goodison Park.
With a quarter of an hour remaining in a scrappy match on Merseyside, USA international Dempsey hit a long-range effort off Sylvain Distin that looped in over Tim Howard.
But as 90 minutes ticked past, a fine header from Steven Pienaar levelled for the hosts, before Nikica Jelavic scored a poacher’s finish at the death to put the Toffees fourth ahead of Spurs on goal difference.
Everton had arguably edged the match in terms of chances, but Spurs – led by Moussa Dembele – had the better of the passing stats and created some decent chances.
David Moyes’s side welcomed back Kevin Mirallas and Seamus Coleman after several games out with injury, while Darron Gibson stood in for skipper Phil Neville.
Spurs meanwhile were missing talismanic winger Gareth Bale, out with a hamstring injury, with a formation switch seeing Emmanuel Adebayor partner Jermain Defoe up front.
Genuine chances were mostly limited by effective defending, the best of them falling to Everton as Hugo Lloris made an excellent stop at the feet of Jelavic, while Steven Caulker blocked a goal-bound finish from Leon Osman.
Boos rang out after the half-time whistle was blown, although they were misdirected at referee Kevin Friend, with replays showing he was correct to reject two penalty appeals from Everton late in the half - Dempsey took a Jan Vertonghen clearance on the chest while William Gallas kept his arms by his side when a Pienaar shot spun up into his midriff.
There was a half-time change for the Toffees as Mirallas was replaced by Steven Naismith, a precautionary measure with the Belgian feeling the hamstring that had kept him out for four matches.
Naismith’s first act was to put a close-range reaction finish wide after a low follow-up drive from Leighton Baines, whose free-kick had been blocked, while Jelavic had an effort blocked by a flying Gallas as Everton started the second half with intent.
Spurs responded with a spell of possession that yielded a string of corners and a Vertonghen free-kick that Tim Howard tipped over.
Chances were at a premium after that initial burst of activity, Osman firing one effort well over as neither keeper was tested around the hour mark.
The hosts were not helped by an ineffectual performance by usual danger-man Marouane Fellaini, who struggled to make an impact as Sandro hassled and harried him in midfield, with Gallas and Caulker largely managing his aerial threat up front.
The game was starting to get punctuated by niggly fouls, and was begging for a stroke of luck or lapse in concentration by one of the two well-marshalled defences.
A lapse did come as a poor clearance by Gallas landed to Osman, but Lloris did well to save his low drive; it was the stroke of luck that broke the deadlock as a long-range shot from Dempsey took a wicked deflection off Distin, leaving Howard for dead as it diverted into the goal.
Spurs then went defensive as they sought to resist the home side’s late rally, but they almost doubled their lead when Gylfi Sigurdsson hit the bar after a counter attack.
But the visitors paid the price for beckoning Everton as, after a fine run and cross from Coleman, Pienaar flew in with a rare header that flew past an unsighted Lloris and into the bottom right.
Everton were not content with the point, piling men forward as the fourth official signalled at least four minutes of stoppage time.
Their bravery was rewarded as, after an attempted bicycle kick from substitute Vellios dropped on the edge of the six-yard box, Jelavic reacted quickest to stroke the finish past Lloris.
MAN OF THE MATCH
Seamus Coleman (Everton): Had a fine game on his return from injury, causing endless problems with his attacking runs down the right, and getting the assist for Pienaar’s goal with an excellent cross.
EVERTON: Howard 6, Coleman 8, Baines 6, Jagielka 7, Distin 7, Osman 6, Pienaar 7, Gibson 6, Fellaini 6, Mirallas 7, Jelavic 7; Subs: Heitinga N/A, Naismith 6, Vellios N/A
TOTTENHAM: Lloris 7, Walker 6, Gallas 7, Caulker 7, Vertonghen 6, Lennon 6, Sandro 7, Dembele 7, Dempsey 7, Adebayor 6, Defoe 6; Subs: Huddlestone 6, Falque N/A, Sigurdsson 7