The visitors dominated the first half and although they created a host of scoring chances before the break they were unable to make the breakthrough.
"It was very tight, a good game. We had good moments and a great first half with pressing and good attacking combinations and we should have gone ahead," Spurs manager Andre Villas-Boas told the BBC.
"The second half evened itself out and in the last 30 minutes they had a go at us. We could have gone second but, considering Everton's aspirations, it was important to get a point."
Everton manager Roberto Martinez told Sky Sports: "Spurs are a very good side and work very hard when they lose the ball and that upset our rhythm.
"But we found that rhythm in the second half and it became an open game with two teams who wanted to win and not settle for a point."
Although Everton rallied after halftime, apart from a cross shot from former Spurs man Steven Pienaar that went narrowly wide of the far post they rarely threatened Hugo Lloris's goal.
The France goalkeeper stayed on the pitch until the end despite taking an accidental knee in the head from Everton striker Romelu Lukaku in the latter stages.
Lloris's injury prompted a lengthy delay and the addition of nine minutes of stoppage time.
Villas-Boas said: "Hugo still doesn't remember the incident with Lukaku. He lost consciousness there but he seemed assertive and determined to continue and showed great character and personality.
"We decided to keep him on based on that."
In the day's other match Cardiff City were playing Swansea City, the first ever Welsh derby in English football's top flight.
(Editing by Tony Jimenez)
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