The visitors, who have not won away against the Toffees 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 that they could have won.
Ultimately they had to settle for a share of the spoils, meaning that Tottenham move up to fourth and Everton, who dropped points at home for just the second time in 11 matches, stay seventh.
|VIEW FROM GOODISON PARK
|By Gary Maiden
Both Everton and Spurs passed up on the chance to climb to second in the table after a goalless draw lacking in quality was played out on Merseyside.
A point for Spurs sees the London outfit climb back into the top four while the Toffees extended their long unbeaten record to 14 games in 2013.
The visitors will perhaps be disappointed not to have wrapped the game up early on as Everton struggled to get out of their own half but the hosts grew back into the game late on.
A draw was probably the right result in the end.
Andre Villas-Boas named an unchanged team from the 1-0 league victory over Hull City, meaning that 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 area 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.
|10/1||Tottenham are 10/1 with Bet365 to beat Sheriff 1-0 in the Europa League|
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 that 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.