Martin Jol opted for three central midfielders to combat the robustness of Everton so Jermaine Jenas returned and Steed Malbranque dropped to the bench.
Dimitar Berbatov gave Tottenham the lead with his first away goal in the Premiership.
"You just think for a second or two and the thought going through my mind was to shoot instantly," he said. "I did that and fortunately it was a great shot to make it 1-0."
However, just over three minutes before the break Everton were back on level terms.
At the break David Moyes pulled off striker Anichebe and replaced him with Gary Naysmith to try and gain a foothold in the midfield battle and gradually Everton took control.
It forced Tottenham's strikers back in support and this had an effect on future striking opportunities. So Jol made a change of his own, replacing Aaron Lennon with Malbranque for the remaining ten minutes or so.
Just as the contest looked like it would peter out as a draw, Jenas scored his fifth goal of the season, his second in as many games.
Chris Hugton said: "On the balance of play I think we deserved to win although going into the final stages it looked set for a draw. We had chances with Dimitar and Robbie and it looked like it needed something special to win it."
"JJ showed that extra quality in his game and delivered with that wonderful strike. He scored a fine goal at Sheffield United where he broke from midfield, but it was a different type of goal this time. We played very well and restricted them to minimal chances."
Martin Jol said: "In the first half they had a couple of situations from the long ball and one good chance with a shot on goal but overall we took control without creating a lot."
"Of course it helped that Berbatov scored after a great attack from Lennon and that's what we want from the little man, to influence away games and he did that with the first goal.
"The second half was different. We couldn't do the same, they took control, changed their shape but it wasn't a big problem because they didn't create a lot. Having said that, Osman had two opportunities but that's why we have Paul Robinson in goal and he saved us.
David Moyes said: "I was delighted to be going in at half-time at one-each because I thought Tottenham had out-played us."
"After finding it difficult to get close to Tottenham in the first half we had worked hard to get back into the game and I thought we did much better in the second half but to lose a goal in the last minute was really disappointing."
Jol added: "One point would have been okay but Goodison has been a good ground for us and I thought we really went for the winner in the last 10 minutes with Robbie Keane, Berbatov and then JJ's goal.
"Overall, I think we were just about the better side."
Michael Dawson said: "It was nice that the game came so quick after winning 4-0 away from home and we were able to take that into the Everton game and put on a good performance, in the first half especially.
"It was disappointing to go in level but we came back out believing in ourselves, Robbo made a couple of great saves and it was a great goal from JJ."
Source: TottenhamHotspur.com/EvertonFC.com