Villas-Boas had been critical of Spurs' support in Sunday's 1-0 defeat of Hull, but on Wednesday night they were in good voice during a thrilling fourth round meeting at White Hart Lane.
Gylfi Sigurdsson put Spurs ahead with an early contender for goal of the season, but Brad Friedel's own goal took the tie to extra-time.
Friedel is mobbed by Kyle Walker and Harry Kane after saving Elmohamady's penalty
Paul McShane then fired the away side ahead, but Harry Kane levelled the scores before Tottenham won a marathon shoot-out 8-7.
Spurs fans sang throughout the match and there were also several chants of "AVB's blue and white army", even though the home side were on the back foot for the majority of the second half.
"It was a great, entertaining game and the supporters were absolutely fantastic," the Tottenham manager said.
Spurs beat Hull
"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."
Aaron McLean put Tottenham on course for victory in the shoot-out when he missed Hull's first penalty, but Erik Lamela failed to score, extending the showdown beyond the original five takers.
Friedel then put the seal on the win when he saved Ahmed Elmohamady's effort from 12 yards.