Mauricio Pochettino's side were joined in the semi-finals by Fulham, who beat Everton on penalties, and second-tier Middlesbrough, 3-0 winners against Port Vale.
Beaten three times in their previous five games, a wretched run that started with their 4-1 loss at Newcastle, Chelsea showed some much needed fighting spirit to win the quarter-final 4-2 on penalties after a 1-1 draw at Stamford Bridge.
Languishing in 11th place in the Premier League, they were two minutes from being knocked out before last-gasp equaliser Ukraine winger Mudryk came to the rescue with the leveller that set the stage for the Blues' shoot-out success.