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."
He added: "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 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."
Stoke boss Tony Pulis blamed Adam's dismissal for his side's failure to mark their 150th anniversary celebrations with a home win, saying: "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."
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."