The tie headed for extra time after the two teams played out a 2-2 draw in 90 minutes, but an extra-time double from Cesc Fabregas saw Chelsea complete a remarkable turnaround and book their spot in the fourth round.
Both managers made seven changes from their respective Premier League lineups at the weekend, with Leicester boss Claudio Ranieri handing starts to the likes of Ben Chilwell, Demarai Gray, Jeffrey Schlupp, Marcin Wasilewski and Ahmed Musa in a much-changed XI.
Chelsea boss Antonio Conte, meanwhile, also shuffled his pack as Asmir Begovic, Fabregas, Victor Moses, Pedro, Ruben Loftus-Cheek and Michy Batshuayi all came into the XI and there was a debut at left-back for deadline-day signing Marcos Alonso.
The visitors had the ball in the back of the Leicester net in the fifth minute when Pedro collected a sweeping David Luiz pass before rounding Ron-Robert Zieler and finishing, but the offside flag was raised as Chelsea's celebrations were cut short.
Both teams had strong penalty shouts inside the opening 15 minutes of action, but the referee first turned down Chelsea's call for a handball from Wasilewski, before the official waved away a Gary Cahill challenge on Musa inside the Leicester box.
Chelsea looked the more threatening in the early stages, but it was Leicester that made the breakthrough in the 17th minute when Shinji Okazaki headed home from close range after the visitors had failed to deal with a cross from Musa, which caused all sorts of trouble inside the box.
Batshuayi should have equalised for Chelsea in the 26th minute when he met a deep cross from Pedro, but the Belgian international headed wide of the post as Leicester remained ahead inside an atmospheric King Power Stadium.
Leicester were always a threat down the other end, however, and scored their second of the night in the 34th minute when King chipped a clever pass in to Okazaki and the striker, courtesy of a bounce, found a route past Begovic for the second time.
The home side were cruising in the latter stages of the first period, but they switched off from a corner in the second minute of stoppage time, which allowed Cahill to find the back of the net with a header to hand Chelsea a lifeline.
Chelsea had a half-chance immediately from kickoff in the second period as Moses raced down the right before setting up Fabregas, but the Spaniard's curling effort was straight at Zieler in the Leicester net, before Schlupp had a low strike deflected just wide down the other end.
The visitors were level in the 49th minute, however, when Cesar Azpilicueta found the top corner with a stunning effort from outside the box to register his first goal since January.
The match settled after a fast start to the second half, although both teams were still threatening as each continued to commit players in the final third of the field. Indeed, Batshuayi came close to scoring Chelsea's third 20 minutes from time, but the Belgian's powerful effort brought a wonderful save from Zieler.
Diego Costa then had a golden chance for Chelsea shortly after coming off the bench, but the Spain international sent his low effort just wide of the post after taking advantage of a mistake from Leicester's defence.
Leicester had two excellent chances to score a third 15 minutes from time, but Begovic managed to produce a fingertip save to keep out an effort from Musa, before King, who was unmarked and just five yards out, somehow headed over the crossbar from the resulting corner.
Costa was causing Leicester's back four all sorts of problems and had two more chances five minutes before the end of normal time, but the striker could not find a route past Zieler, who stood strong under the pressure.
Leicester were reduced to 10 men in the final minute of normal time when Wasilewski was handed his second yellow card for an elbow on Costa, but the home side managed to hold off some late Chelsea pressure to take the match into extra time.
Chelsea scored a third just two minutes into extra time, however, as Eden Hazard set up Fabregas and the Spaniard gave Zieler the eyes before placing his effort into the back of the net from inside the box.
Two minutes later, it was game, set and match as Fabregas smashed the ball home from close range and that proved the end of the scoring at the King Power Stadium as Conte's side returned to winning ways.
Leicester will attempt to bounce back from the defeat when they visit Manchester United in the Premier League on Saturday, while Chelsea will travel to Arsenal on the same day.