Players revel in penalty win
Moyes was fuming with his side following their 2-0 defeat to Bolton Wanderers last weekend, describing the performance as "as bad as I can remember since I've been in charge".
However, after Leighton Baines had responded to Frank Lampard's goal in extra-time with a fine free-kick with just a minute remaining, Everton defeated the FA Cup holders 4-3 on penalties at Stamford Bridge.
"It was questioned last week [Everton's work-rate] and I said the players here normally respond, and they certainly responded today," Moyes told ESPN.
"The games against Chelsea have been really tight over the years - I actually thought in the first game we played really well and but for a save from Cech and a great tackle from Essien we would have been two up and probably through to the round, so to come back to Stamford Bridge again this season was a daunting task to a side that has not lost a game in the cup for how long, two or three years now?
"It doesn't matter. The boys stuck at it, we got a free-kick at the end and it was a great goal and obviously the penalty kicks were excellent.
"It was postage-stamp [from Baines], because the goalkeeper is very hard to get the ball past. But you know something, it was no more than we deserved over the two games. We rode our luck with a couple of things and in the second half Chelsea got much better, but no more than we deserved in the game, the players were terrific the whole game."
The defeat of Chelsea was welcome relief for an Everton side currently sitting in 13th place in the Premier League, just three points clear of the relegation zone.
Moyes has been under increasing pressure of late and would not speculate whether victory could help transform Everton's season. He said: "I have been asked that question every week for 25 weeks now so all I can say is we are through in the cup.
"You saw an Everton team and an Everton performance with a great team spirit, and you saw a great deal of good ability as well, they are really good players."
Opposite number Carlo Ancelotti was left to rue another day of missed chances for his Chelsea side, who are in real danger of their worst finish to a season since Roman Abramovich bought the club.
"We deserved to win,'' said Ancelotti, whose side now must beat FC Copenhagen in the last 16 of the Champions League to stand any chance of ending the season with silverware. "Obviously, we are not happy but we have to look forward, to use these days, the next days, to recover well, to recover energy and to prepare well (for) the next game in the Champions League. I think that we have the time. It will not be easy but we have the time.''
Ancelotti refused to blame his penalty-takers and insisted he did not see a nudge by John Heitinga on Cole as the latter walked forward to take his ill-fated spot-kick.
"We had very good shooters from the penalty. Lampard, [Didier] Drogba, Anelka, Essien and Ashley Cole: they were the best,'' Ancelotti said. "Obviously, penalties are a lottery and sometimes you can win and sometimes you can lose.''
Ancelotti insisted Chelsea should have had a spot-kick in normal time when Ramires was booked for diving and admitted Branislav Ivanovic might have done better in conceding the Baines free-kick.
With the Premier League title all but surrendered and the champions facing a fight even to finish in the top four, Ancelotti is under more pressure than ever. But he said: "This is football. Every time the team doesn't reach the result, obviously the manager is under pressure. I have fantastic support from the club, from my players. And together we have to move on.''