First-half goals from Gareth Bale and Aaron Lennon, before Per Mertesacker responded early in the second half, allowed third-placed Spurs to move seven points clear of the Gunners but Villas-Boas was not getting carried away with the result.
"It was very, very important for what it means, but obviously nothing is finished yet," he told Sky Sports.
"We have to believe that we still have to fight hard. We have to play Liverpool (next weekend) and we have to try to profit from another important fixture and keep doing our job.
"It is important we gather as many points as we can to finish where we want to be."
Villas-Boas touched on the fact that Arsenal were able to bridge the gap of seven points last season and eventually finish above Tottenham.
"Last year, by this time, Arsenal made the difference of seven points to Tottenham and we know how it finished," Villas-Boas added.
"Motivations are now different. We are extremely confident. They are on a low at the moment.
"We have to continue to do our job."