Daniel Sturridge put the Londoners into the quarter-finals with a winner after 116 minutes of a remarkable tie at Goodison Park in which both sides had a man sent off and missed a penalty.
Villas-Boas, whose first-choice side were reduced to nine men last weekend, said: "We were very committed, we showed resilience and strength of character once again."
He added: "We went down to 10 men, unfortunately, once again and made things even more difficult but we were able to triumph in difficulty and these are good signs for us.
"We got one of the most difficult draws, Everton away with one of their strongest sides. They showed they wanted to go through this phase but we were able, not only to play good football, but show good commitment at desire."
Everton boss David Moyes felt the result was harsh on his team but was not looking for excuses.
The Scot said: "I don't think we deserved to lose. Certainly over 90 minutes we should have won the game. We have only got ourselves to blame for not winning it in 90 minutes.
"It is one of those things that happens. Someone has got to win, someone has got to lose and we lost this time.
"When it was 11 v 10 I thought the worst we were going to get was a penalty shoot-out.
"When it went to 90 minutes, both teams looked really tired but I thought the boys did well."
Source: PA
Source: PA