"It was difficult," said Villas-Boas, who made five changes to his side. "At 2-0 Leeds got further inspiration and we wanted to get back quickly.
"We had a good 15 minutes before the goal when we had some good chances, and I think it was important to score early.
"It was difficult as we were always chasing the game and this is a difficult place to play, and we couldn't play our football as well as we could have.
"We wanted to go through in the competition, but we were knocked out by a Leeds team who were extremely competitive today [Sunday], and you have to give credit to them."
Tottenham were without Emmanuel Adebayor, on duty for Togo in the African Nations Cup, while Villas-Boas confirmed absent striker Jermain Defoe had injured himself in training.
But the Spurs boss refused to comment on whether he would be drafting in any forward cover in the final week of the transfer window, and did not accept his side would benefit from Sunday's defeat in terms of their bid for Champions League qualification.
Neil Warnock took a swipe at the Leeds boo-boys after his side incredible win.
"I'll probably open a bottle of champagne tonight [Sunday] and have a few little tipples," said Warnock, whose side were criticised by fans after the recent league defeat at Barnsley.
"I'm sure I'll be able to do something like that. I might even treat myself to a bag of crisps."
Some Leeds fans are angry at the expected departure of leading goalscorer Luciano Becchio, but Warnock said Sunday's display should help lift the gloom.
"We've had negative press on some of the results lately and yet we've won six on the trot at home (now seven)," he said.
"But in Leeds the glass is always half empty, it's never half full in Leeds. That might be a Yorkshire trait."
Warnock rated Sunday's win among the best since he joined the club in February last year.
"It's a fabulous day for everybody," he said. "Not just because we won the game, but we had a few problems a few weeks ago, there was a little bit of dissent around the place.
"We go to Birmingham and nobody expected us to win and 1,500 fans followed us, and today [Sunday] was a thank you to them because they got behind us and we ended up winning that game, which gave us the opportunity to play Spurs.
"I really enjoyed the match. We didn't get anything other than what we deserved today [Sunday] and we had a few chances as well.
"It was a typical FA Cup game. They were like that when I was a kid.
"They played a strong team. To see Bale, Lennon and Parker in that team and to beat them is fabulous because we had a very young team."