"It was very, very important for what it means, but obviously nothing is finished yet," Villas-Boas told Sky Sports.
"We have to believe that we still have to fight hard. Arsenal have to play Man City, we have to play Liverpool (next weekend).
"We 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.
"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."
For Arsenal and its beleaguered manager Arsene Wenger, the gap shows no sign of narrowing any time soon.
For 16 years it has been the ruler of its own backyard, but the neighbor is beginning to stir.
Without a trophy in eight years, on the brink of an exit from the Champions League, the Frenchman is facing the most difficult spell since he arrived at the club in October 1996.
But Wenger refutes any suggestion that Tottenham has overtaken his side in terms of quality.
"That's not the impression I got from the game today," he told Sky Sports News. "We were not efficient in those decisive zones, not at the back or up front.
"We were not cautious enough. On the two goals we played offside in a position where we shouldn't play offside.
"It is difficult to struggle with defeat because of the effort we put in and the energy level we put in over the 90 minutes was tremendous.
"We made it difficult for ourselves. We came back and dominated the game but the game was stop and go.
"It was difficult to get a flow in the final 20 minutes, we didn't find the second goal."
But as fans got back onto the London Underground this evening following the contest, the question marks over Wenger's future remain.
Monday will mark his 6000th day in charge and while supporters may be losing faith, Wenger is adamant Arsenal can still qualify for next season's Champions League
He added: "It's very frustrating, but we have to put the energy in every game like we did today and keep going, you never know.
"It would be absolutely terrible not to fight like mad because there are a lot of positives in the game.
"What happened today we have seen repeated so many times in big games that of course it's difficult."
In Italy, Rodrigo Palacio spared Inter Milan any embarrassment with a late cameo performance to steal a 3-2 win at Catania.
Trailing 2-0 at the break thanks to strikes from Gonzalo Bergessio and Giovanni Marchese, Inter fought back in the second half to turn the game on its head.
Ricky Alvarez halved the deficit before Palacio fired home an equalizer with 20 minutes remaining.
And with the contest entering the second minute of stoppage time, Palacio found space to drill home a dramatic winner.
"We had a terrible 20 minute period and were punished for our mistake on the first goal," coach Andrea Stramaccioni told Sky Italia.
"Luckily this team has thick skin and managed to fight their way back into the game.