The former Barcelona winger capitalised on some poor goalkeeping from Kasper Schmeichel to nod home N'Golo Kante's cross in the 105th minute and end the Foxes' hopes of a place at Wembley.
Earlier, Jamie Vardy's second-half equaliser had cancelled out Alvaro Morata's first goal since Boxing Day to send an entertaining tie beyond the 90 minutes.
Although Chelsea boss Antonio Conte said beforehand the FA Cup was "not important" to him on a personal level, it now represents the best chance of silverware for the soon-to-be dethroned Premier League champions and his strong starting XI reflected that.
Despite that it was Leicester who started the better, Marc Albrighton firing Danny Simpson’s right-wing cross narrowly past the right-hand post.
But Chelsea gradually found a foothold, thanks largely to Kante’s intelligent promptings against his former club while wing-backs Victor Moses and Marcos Alonso were finding plenty of space down the flanks.
Wilfred Ndidi forced Willy Caballero into a fine save down to his right after 20 minutes but the visitors looked the more assured in possession.
Morata had already fired straight at Schmeichel from the edge of the box before he finally ended his goal drought three minutes before the break.
Willian led a swift counter attack before slipping a delightful through ball to the former Juventus man and he showed a cool head to clip past Schmeichel and into the net.
Leicester came out strongly after the break and Riyad Mahrez could have levelled within two minutes, his 45-yard attempt following a poor clearance from Caballero drifting wide of goal.
Vardy then skewed a header off-target when completely unmarked eight yards out but he was not to be denied with 15 minutes to go.
Mahrez was the architect with a run and cross down the right flank that caused chaos inside Chelsea's six-yard box. Vardy and Vicente Iborra saw shots blocked before the latter forced another effort on goal. Caballero made a smart save low to his left but the loose ball fell nicely for Vardy to power into the net.
Both sides went for a winner in the final 10 minutes - Morata backheeling against the post from a Cesc Fabregas cross, although the offside flag was up, and Vardy firing low at Caballero before Schmeichel was quickly off his line to deny Morata.
The extra period started slowly and it came as a surprise when Chelsea went back in front. Kante's cross into the box did not look overly threatening but Schmeichel's decision to charge off his line proved foolhardy as Pedro simply nodded over him and into the empty net.
Leicester pushed for a second leveller - Moses producing a sensational block to thwart Fousseni Diabate - while, at the other end, Albrighton denied Kante in similar circumstances.
Key Opta stats:
- Chelsea have reached the FA Cup semi-finals for the second successive year, and for the fourth time in the last seven seasons.
- Of the last three occasions where Chelsea have faced Leicester in the FA Cup before this season, the Blues have knocked them out en route to winning the competition (1997, 2000 and 2012).
- Leicester have now lost back-to-back FA Cup quarter-final matches, with both defeats coming against Chelsea. In fact, the Foxes have lost each of their last four FA Cup quarter-finals, failing to reach the Semi-finals since 1982.
- Excluding his defeat in last season’s final against Arsenal, Antonio Conte has now progressed from all nine of his FA Cup ties in charge of Chelsea.
- Jamie Vardy has now netted 10 goals in his 12 appearances in all competitions against ‘big six’ opposition this season.