Earlier the home team's England international Jamie Vardy had equalised following Alvaro Morata's first goal of 2018.
With no replays from the quarter-finals onwards, 30 extra minutes were required, in which Pedro headed in a cross by former Leicester midfielder N'Golo Kante.
Manager Antonio Conte, knowing this is his last chance of a trophy this season, must hope the victory is an omen. The previous three times Chelsea have knocked Leicester out, in 1997, 2000 and 2012, they won the competition.
In the semi-final at Wembley they will face Southampton, the lowest placed of the four teams left.
Morata finished well to open the scoring just before halftime after Riyad Mahrez lost possession and Willian instigated a swift counter-attack and played the striker in with a perfect through pass.
Left out of Spain's squad for two forthcoming friendlies after a run of poor form, the 58 million pound ($US81 million) signing from Real Madrid had not been on the score-sheet since 26 Dec.
In contrast it was Vardy's seventh goal in nine games in all competitions. After a spell of pressure, Mahrez put in a low cross and although Vicente Iborra was twice denied by blocks, Vardy clipped home the loose ball.
Chelsea could have repeated their goal when Willian put Morata through again, Kasper Schmeichel bringing off a double save.
In extra time, substitute Pedro missed for Chelsea but made amends when the Leicester goalkeeper was caught in no-man's land.
"Leicester caused us lots of problems but I think we did very well," said man-of-the-match Kante.
Manager Conte was delighted with his team's performance after being knocked out of the Champions League by Barcelona in midweek.
"A tough game but we knew this," he said. "We must be pleased today. I saw a good performance and great character from my team."
The win ended a run of five away matches without success and Chelsea will now be favourites to reach another FA Cup final.
Losing the final to London rivals Arsenal last season cost them a Premier League and FA Cup double but they have won the trophy seven times, the last four of them between 2007 and 2012.
Leicester manager Claude Puel felt his team deserved better and declined to blame Schmeichel.
"I can't say anything wrong about him - he's a fantastic keeper," he told the BBC.
