Spurs won a marathon shoot-out 8-7 when Brad Friedel saved Ahmed Elmohamady's effort to advance to the quarter-finals after extra-time finished 2-2 at White Hart Lane.
It was Tottenham's first shoot-out win since 1994, setting up a showdown with London rivals West Ham in the last eight.
"We practiced and practiced penalties for the Basel game last year, but we didn't this time," Villas-Boas said.
"We haven't practiced 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."
Gylfi Sigurdsson put Spurs ahead with a brilliant goal in the 16th minute before Friedel's own goal sent the tie to extra-time.
Paul McShane scored first for the visitors in the additional half hour but Harry Kane levelled the scores to set up a thrilling shootout.
Villas-Boas criticised Tottenham fans following Sunday's Premier League triumph over the Tigers but heaped praise on the home supporters after Wednesday's win.
"It was a great, entertaining game and the supporters were absolutely fantastic," Villas-Boas 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."
Thank you for reading 5 articles this month* Join now for unlimited access
Enjoy your first month for just £1 / $1 / €1
*Read 5 free articles per month without a subscription
Join now for unlimited access
Try first month for just £1 / $1 / €1
