Emmanuel Adebayor gave Spurs a 34-second lead, but strikes from Phil Jagielka and Kevin Mirallas looked set to condemn Villas-Boas' side to a second consecutive home league loss before Gylfi Sigurdsson earned a 2-2 draw late on.
Despite only claiming four points from the last four Premier League games Tottenham remain in the Champions League qualification spots, but Villas-Boas has acknowledged strong displays against City and Chelsea will go a long way to deciding the team's end-of-season achievements.
|AVB OPTIMISTIC OVER BALE INJURY
|11/10||Tottenham are 11/10 with bet365 to finish in the top four
"Credit to Everton because of their organisation and how good they can be. In the end it was important for us to salvage a point against a team that was fighting for the same objectives as us.
"Everton are a strong team and they had good chances. If you give them a small amount of space they punish you and they did ever so well and were a great opponent. It was an entertaining game of football but I don’t agree that we relaxed."
Tottenham were without key man Gareth Bale for the meeting with Everton after the attacker was ruled out for two weeks with an ankle injury, but Villas-Boas does not believe his absence was the sole reason they did not claim all three points.
"I think it is unfair to say we missed Gareth Bale today," the Portuguese added. "We’re not only missing Gareth, but Aaron Lennon and Sandro. This gives chances for other people to step up and we’ve seen them step up their level today; Sigurdsson, [Lewis] Holtby and Adebayor.
"I think we set up to try to win the game and it was difficult for us to accept the second half as we started brightly with intensity."