Spurs failed to make the most of considerable possession and a number of chances in the first half while Everton dominated after the break.
Both sides thought they had strong penalty claims after Jan Vertonghen and Seamus Coleman tangled with each other on separate occasions in each half but nothing was given.
Home goalkeeper Tim Howard and opposite number Hugo Lloris - who played on after a blow to the head in the second half - both had busy afternoons but held firm.
Both managers fielded familiar line-ups. Spurs reverted to the starting XI which beat Hull last weekend, meaning Erik Lamela was back on the bench, while Everton replaced Ross Barkley with Leon Osman in their only alteration from their last league game.
Spurs controlled the game before the break as the hosts struggled to gain and then retain possession.
The Londoners, backed by a vociferous visiting support, had their first sight of goal in the opening minutes as Sandro forced a fine save from Howard.
Andros Townsend then tried his luck but missed the target and Roberto Soldado spurned a gilt-edged chance as he glanced a header wide from an Aaron Lennon cross.
Townsend had another shot deflected over and Howard parried a well-struck free-kick from Kyle Walker as Spurs continued to press.
Vertonghen then went down in the area under a challenge from Coleman but referee Kevin Friend did not even think about awarding a penalty.
Howard saved again from a Lewis Holtby shot and almost half an hour had gone before Everton seriously threatened for the first time. It was Osman who had the opportunity, finding space to shoot and then firing narrowly over Lloris' bar.
Kevin Mirallas had a free-kick deflected behind for a corner as Everton began to find their feet in the game, but still it was Spurs who dominated possession.
Everton began the second period with renewed vigour and looked to use Romelu Lukaku's pace and strength more through the middle.
Lukaku launched one good counter-attack, feeding Belgium team-mate Mirallas to his left but the impressive Vlad Chiriches slid in to deny a sight of goal.
Another good move created space for Steven Pienaar on the edge of the box but Michael Dawson stood tall to block and was winded as he took the full force of the South African's well-struck effort in the stomach.
Everton sent on Barkley for Osman to maintain their momentum and the youngster immediately got into the game as he flashed a drive over.
The Toffees then wanted a penalty when Coleman appeared to have the back of his legs clipped by Vertonghen. Coleman, to his credit, tried to play on and get a shot away but was still furious that Friend did not blow his whistle.
Barkley was influential, suggesting Roberto Martinez's decision to start him on the bench may have had the desired reinvigorating effect, and spread play wide for James McCarthy.
McCarthy whipped in a dangerous cross from the byline and Lukaku glanced across goal.
After a quiet spell, Howard was called back into action to punch away a curling effort from Spurs substitute Gylfi Sigurdsson.
There was immediate concern for Lloris at the other end as he raced out to claim a long ball and collided with Lukaku at full pace.
It looked an accidental clash, with the diving Lloris banging his head on Lukaku's knee, but Friend further alienated himself to fans by booking the Everton forward.
Farce followed as Lloris was led off and Brad Friedel got ready to take his place, but the Frenchman wanted to continue.
He had to leave the field but immediately came back on, and although medical staff still appeared to try to persuade him otherwise, he returned to the game and Friedel sat down.
Lloris seemed to show no ill-effects as he beat away a powerful strike from Gerard Deulofeu.
Chances continued as the game went into nine minutes of injury time and Phil Jagielka blocked bravely from Sigurdsson, who then screwed wide.