Chelsea side, who are in real danger of their worst finish to a season since Roman Abramovich bought the club, lost their fourth-round replay on penalties.
Frank Lampard scored in extra-time to seemingly set his side on their way to victory - but Leighton Baines' scored from a superb free-kick a minute from time to push the game into penalties.
Baines missed his but all other Everton players scored while Nicolas Anelka and Ashley Cole missed theirs to give the Toffees a 4-3 win.
"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.