Spurs were on the back foot for much of the second half and conceded an equaliser when goalkeeper Brad Friedel spilled the ball into his own net, but the home faithful remained in good voice throughout.
"It was a great, entertaining game and the supporters were absolutely fantastic," Sky Sports News quotes Villas-Boas as saying.
"To see their response after we suffered a setback was very pleasant, they can make that difference for us. It was a great atmosphere and it was great for the team to feel that empathy with the supporters."
Spurs will take on West Ham United in the quarter-finals of the competition.