The Foxes appeared to be coasting to victory after two goals from Shinji Okazaki, but Gary Cahill headed in from a corner to halve the deficit before the break.
Moments after the restart, Cesar Azpilicueta scored a long-range volley before two extra-time goals from Cesc Fabregas gave Chelsea a deserved win at the King Power Stadium.
The Italian is quoted by BBC Sport as saying: "I am happy with our performance until 2-0. We started very well and we were in control of the match. Chelsea had more possession but we were very smart and clever to score twice.
"The key was when we conceded a goal on the last corner of the half - we lost two dangerous men, Luiz and Cahill.
"In extra time I said stay together and close the space, but Fabregas scored and it was very difficult for us."
The win for Chelsea was their first in three games in all competitions.