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The Londoners, dumped out of the Champions League as holders and beaten in the Club World Cup final already this month as players and dubious fans adjust to interim boss Rafa Benitez, went a goal down before storming back in heavy rain.
Leeds, big rivals of Chelsea after a series of fractious games between the sides since an infamous 1970 FA Cup final, went ahead on 37 minutes through Luciano Becchio after Chelsea defender David Luiz had charged up field and lost the ball.
Chelsea, who had numerous chances in the first half, hit back through Juan Mata just after the break when his shot from the edge of the box following a fine move foxed the goalkeeper.
Branislav Ivanovic headed in Frank Lampard's corner at the near post to make it 2-1 on 64 minutes and forward Victor Moses soon increased the lead when he fired in after a mazy run.
Eden Hazard then latched onto David Luiz's long pass before Fernando Torres completed the scoring for a strong Chelsea lineup, who host Villa in the Premier League on Sunday lying third.
Swansea City, Villa and fourth tier Bradford City, who stunned Arsenal, had already progressed to the last four last week as they chase a trophy which provides a Europa League berth.