The visitors, who have not won at Everton since 2007, controlled the first half but could not make their chances count, while Everton were a much brighter prospect after the interval.
Referee Kevin Friend waved away strong penalty shouts from either side in an entertaining game that both will feel they could have won.
Ultimately they had to settle for a share of the spoils, meaning Tottenham move up to fourth and Everton, who dropped points at home for just the second time in 11 matches, stay seventh.
Everton left Ross Barkley out of their starting line-up for the first time in the Premier League this season, Leon Osman replacing him in central midfield.
Andre Villas-Boas named an unchanged team from the 1-0 league victory over Hull City, meaning Lewis Holtby was afforded the opportunity to play against the club he supported as a boy.
In-form winger Andros Townsend was granted permission to play with a splint on his injured right wrist.
Tottenham carried the most potent threat early on, Sandro forcing a save from Tim Howard with a swerving long-range strike that almost deceived the American goalkeeper, who tipped it past the post.
Aaron Lennon escaped down the left after five minutes, cutting back onto his right foot to pick out Roberto Soldado, but the Spaniard's header from 10 yards was wayward.
James McCarthy's foul on Jan Vertonghen earned him a booking and handed the visitors another chance to apply pressure, with Kyle Walker's 20-yard effort finding the clenched fists of Howard.
Tottenham saw appeals for a penalty turned down midway through the half when Vertonghen went down under a clumsy looking challenge from Seamus Coleman.
It took until the 28th minute for Everton to trouble Hugo Lloris with anything more than a hoisted ball into the box, and the Frenchman confidently waved Osman's rising shot over the crossbar.
Romelu Lukaku's foray into the box caused a rare moment of panic in Tottenham's defence but Vertonghen tidied up, with Everton beginning to show signs of growing into the game in the last 10 minutes of the half.
The hosts created their most dangerous opening in the first few minutes after the break and Tottenham were indebted to Vlad Chiriches as he made a superb last-ditch tackle on Kevin Mirallas, who had been sent clear down the right by Lukaku.
Barkley replaced Osman with just over an hour gone and the 19-year-old quickly got his first shot away, sending a swerving effort clear of the bar.
Coleman thought he had earned his side a spot-kick in the 65th minute but Friend saw nothing wrong with Vertonghen's tackle just inside the box, despite the Dutchman appearing to make contact with the Irish defender and not the ball.
Lloris looked set to be withdrawn after an accidental clash with Lukaku but, with Brad Friedel ready to come on, the France international appeared to overrule his bench and remained on the pitch.
He was tested with two minutes of normal time remaining and showed no hesitation in racing out to deny substitute Gerard Deulofeu in a frantic finish that included nine minutes of additional time.
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