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.
Chelsea, who had fielded a largely second-string side, went ahead through Salomon Kalou after a dreadful error by Everton back-up goalkeeper Jan Mucha but were pegged back by Louis Saha.
After Chelsea lost their keeper Ross Turnbull for bringing down Saha - an incident that led to Leighton Baines copying Nicolas Anelka by missing a penalty - Everton spurned numerous chances to win in normal time and the visitors edged it after Royston Drenthe was also sent off.
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.
"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.
"It shows we are progressing and it is very satisfying for everybody."
Villas-Boas, who had no complaints about Turnbull's dismissal, admitted his side were hanging on in the closing minutes of normal time.
He added: "Everton had the desire and motivation, they were on top and we had to hold on until extra time arrived.
"What satisfies more is the talent came out in this situation and when the numbers got even they found better spaces and we got the goal we wanted."
Villas-Boas would not answer questions on the ongoing situation involving captain John Terry, who is the subject of a Football Association investigation into allegations of racial abuse.
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."
Moyes refused to criticise Mucha for his first-half blunder in allowing a tame Kalou lob to slip through his fingers into the net.
Moyes said: "I don't think it hard any effect after that. The boy played really well, especially in the second half."