Adebayor poked home a Clint Dempsey cross in the 83rd minute to earn his side a 2-1 win after Dempsey had cancelled out a third minute opener from Steven Nzonzi for Stoke.
Spurs had struggled to fashion a breakthrough despite dominating after Charlie Adams' 47th minute sending-off, but Adebayor's strike lifted them back above north London rivals Arsenal, who play their game in hand against Wigan on Tuesday.
Villas-Boas said: "It keeps us in the frame and gives us the possibility of making the Champions League so the important thing is the dream is still alive for us.
"We wanted to take it to the last day when we are playing at home, so it was an extremely important win for us for those reasons, and hopefully someone else will slip up. Arsenal are strong and experienced but it will go down to the wire."
Villas-Boas singled out Dempsey for special praise after the American scored a superb 20th minute equaliser to cancel out Nzonzi's early strike, then played a crucial role in Adebayor's crucial winner.
Villas Boas said: "He (Dempsey) has been absolutely immense this season with his assisting and his goalscoring and he has sometimes not got the credit he deserves.
"He is a hard worker and a very complete athlete and he had an amazing game. It was an extremely good individual performance.
"We came back from difficult circumstances because part of Stoke's strength is their delivery at set-plays and the delivery (for the goal) was absolutely amazing.
"But we managed to control the other ones and the circumstances turned for us. I am extremely happy for the team in the way we were patient in the game, we kept the ball moving and eventually we got the breakthrough."
Stoke boss Tony Pulis blamed Adam's dismissal for his side's failure to mark their 150th anniversary celebrations with a home win.
Pulis challenged both bookings which led to Adam's sending off and claimed his departure had ruined the game as a spectacle.
He said: "I don't think Charlie has touched (Jan) Vertonghen on the first one and on the second one he's made a mistake by sliding in but again I don't think he touches him.
"It's disappointing because I watch football day in, day out and you see some challenges that don't even get bookings, then you see two challenges like that one today getting a sending-off.
"What disappoints me is there was a full house here for a great day to watch a really good game of football and it has just been taken away from them. The game was then not a game.
"Good teams will frustrate the life out of you and what you mustn't do is get dragged into it and lose your shape and your discipline.
"We're disappointed in particular with their second goal because it was a deflection that let their lad in. That was the little bit of luck we needed that maybe we didn't get today."
Despite the Potters' defeat, their survival in the top-flight is secure for another season, and Pulis added: "I'm delighted for us to be in the Premier League for a sixth consecutive season.
"It's not just what the club have done on the pitch, it's what the club has achieved behind the scenes as well."