Spurs were 2-1 winners over West Ham, while Chelsea needed two late goals to see off Brentford 2-0
All three matches went ahead in front of full crowds despite the worrying rise of coronavirus case numbers that had seen last weekend's Premier League schedule decimated by call offs.
Leicester's previous two fixtures had been called off due to a combination of a coronavirus outbreak in the squad and a mounting injury list.
However, Brendan Rodgers was able to name a close to full strength side on his return to Anfield and it showed early on against a much-changed Liverpool.
Jamie Vardy struck twice inside the first 13 minutes as he angled a drive across Caoimhin Kelleher and then tapped in Patson Daka's pass from close range.
Jurgen Klopp had made 10 changes from the side that drew 2-2 at Tottenham on Sunday with Liverpool also depleted by four positive Covid cases.
However, it was one of their more experienced starters who got them back in the game as Alex Oxlade-Chamberlain smashed home from the edge of the area.
Leicester restored their two-goal cushion before half-time thanks to James Maddison's piledriver from 30 yards that flew past Kelleher.
Klopp introduced three first-team regulars at half-time and one of them, Diogo Jota, gave Liverpool life again 22 minutes from time.
Kasper Schmeichel then needed to produce a stunning save to deny the Portuguese a second, but Leicester still could not hold out as Takumi Minamino sent the game to penalties in the fifth minute of stoppage time.
Minamino then missed with the chance to win the tie from the spot in the shootout, but Kelleher saved from Luke Thomas and Ryan Bertrand before Jota fired home the winning penalty.
Bergwijn ends goal drought
Tottenham boss Antonio Conte showed his determination to end the club's 13-year trophy drought as soon as possible as he named Harry Kane among a strong starting line-up.
But it was the other two of Spurs' front three who were on target in north London.
Steven Bergwijn scored his first goal of the season after two clever one-twos with Pierre Emile Hojbjerg.
Tottenham's lead lasted just three minutes as West Ham seized upon Eric Dier's slack pass and Jarrod Bowen fired low past Hugo Lloris.
Two minutes later it was 2-1 as Bergwijn this time turned provider for Moura to prod home from close range.
West Ham came agonisingly close to an equaliser in stoppage time when Andriy Yarmolenko's deflected effort came back off the bar.
Chelsea boss Thomas Tuchel handed debuts to academy prospects Harvey Vale, Jude Soonsup-Bell and Xavier Simons with the Blues short of seven players due to a Covid outbreak.
However, Tuchel emptied his bench in the second half with the introduction of N'Golo Kante, Mason Mount, Christian Pulisic, Reece James and Jorginho.
And that extra quality made the difference as Pontus Jansson turned James' cross into his own net 10 minutes from time.
Jorginho then stroked home a penalty after Pulisic had been upended in the box.
Arsenal make up the semi-finalists after they thrashed Sunderland 5-1 on Tuesday.