Spurs needed a late Gylfi Sigurdsson equaliser to pick up a point after goals from Phil Jagielka and Kevin Mirallas for Everton had overturned Emanuel Adebayor's early opener.
The result leaves Spurs fourth, just behind Chelsea on goal difference, while Arsenal trail by two points and Everton are four further back.
Tottenham have played a game more than all their rivals, and Villas-Boas said: "It's going to go down to the wire as it stands with Arsenal, Everton, Chelsea and Tottenham all in the frame for that Champions League qualification.
"In games like this you can really see how much desire is in both teams to get those objectives."
He added: "The positive aspect is we've been in this position, chasing, before. Arsenal feel their belief is because they finish seasons very strongly and that belief helps them.
"We don't look at ghosts in the past. We try to do our job properly. We want to win games."
The game was played on Sunday due to Spurs' ongoing participation in the Europa League and Villas-Boas was happy with his side's effort after a gruelling 2-2 draw against Basel on Thursday night.
"I think it was a good game, full of opportunities for both sides, a really competitive match," he added.
"We wanted to win so obviously we're not pleased but bearing in mind the circumstances, bearing in mind our super effort on Thursday it's good to see the team react once again.
"We left it late which didn't give us a chance to go on for the third but I think that their second goal comes a little bit against the run of play because we started the second half very, very well.
"But credit to an Everton side which battled their way through and played for an entertaining game."