Stoke, celebrating the club’s 150th anniversary, took a first-half lead when Steven Nzonzi nodded past Hugo Lloris from a Charlie Adam free-kick.
However, Clint Dempsey levelled in the 20th minute when the American guided a finish over Stoke goalkeeper Asmir Begovic, who was caught of position.
The Potters were reduced to 10 men at the start of the second half after Adam was send off for a second bookable offence, and Adebayor continued his goal-scoring run with the winner in the 83rd minute.
“What we did was we kept the patience. It would have been very easy for us to lump it forward. We were rewarded in the end for that patience. In the end we got what we deserved,” Villas-Boas told BBC Sport.
“It is not easy – we had to be very patient but in any set play Stoke would have a chance to score again.
“It was difficult to break them down.”
“This puts more pressure on Arsenal but they are able to cope with it. We hope Wigan have their day.”