The Blues opened the scoring through Willian after just five minutes but Harry Kane pulled Spurs level in the 18th minute.
Willian restored Chelsea’s lead just before half-time from the penalty spot. But seven minutes into the second half Spurs equalised again, this time Dele Alli adding his name to the scoresheet.
However, Antonio Conte’s Premier League leaders rallied in the final 15 minutes and romped to victory, first Eden Hazard putting his side ahead again before Nemanja Matic made sure, making it 4-2 in the 80th minute from Hazard’s assist.
Chelsea will play the winner between Arsenal and Manchester City, who face off in the other semi-final on Sunday.