Villas-Boas had been critical of Tottenham's home support in Sunday's 1-0 win over Hull, but on Wednesday night they were in good voice during a thrilling fourth-round meeting against the Tigers in north London.
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.
"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.
It was Tottenham's first shoot-out win since 1994, and the north London club will now play a grudge match against West Ham, who won 3-0 at White Hart Lane three weeks ago, in the quarter-finals.
"We practised and practised penalties for the Basle game last year, but we didn't this time," Villas-Boas revealed.
"We haven't practised them for three or four months.
"We've seen some brilliant, brilliant football tonight. If you want to see predictable endings or happy endings you go to the opera or the theatre. In football you don't know what is going to happen.
"I couldn't be happier."