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.
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 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."