Liverpool booked their spot in the semi-finals after edging Leicester City 5-4 on penalties following a 3-3 draw in regular time at Anfield.
The Reds made wholesale changes, as only Tyler Morton retained his place from the thrilling draw with Tottenham Hotspur, with most of the regulars rested while Leicester named a strong line-up, with Jamie Barry among those recalled.
Vardy scored a quick brace to give Leicester a two-goal lead in the first 13 minutes as he twice went through on goal to fire strikes across and past goalkeeper Caoimhin Kelleher.
Alex Oxlade-Chamberlain pulled a goal back for Liverpool six minutes later with a strike from the edge of the box but James Maddison restored a two-goal cushion for City in the 33rd minute with a powerful strike into the roof of the net.
Second-half substitute Diogo Jota pulled a goal back for the hosts 22 minutes before time after he was setup by Takumi Minamino and Minamino smashed home a dramatic equaliser for the Reds deep in stoppages.
The match went straight to penalty shootouts and Jota scored the winning penalty to clinch a 5-4 win for Liverpool.
Chelsea progressed to the semi-finals with a 2-0 win over Brentford at Brentford Community Stadium.
The Blues heavily rotated the side, with only Trevoh Chalobah, Cesar Azpilicueta and Marcos Alonso retaining their starting berths while Brentford kept the core of their regular line-up.
The Bees had the best chances in the first stanza, with Yoane Wissa coming the closest with a free header into the hands of goalkeeper Kepa.
At half-time, Thomas Tuchel brought on Jorginho and Christian Pulisic in a double change, with the likes of N'Golo Kante also getting a run late in the game.
The Blues had much more dominance from then on as they broke the deadlock ten minutes before time after Pontus Jansson cleared Reece James' cross into his own net and Jorginho sealed the deal with a penalty five minutes before time.
Tottenham Hotspur booked their spot in the semi-finals after edging West Ham United 2-1 at the Tottenham Hotspur Stadium.
Spurs made a number of changes to the side that drew with Liverpool in a four-goal thriller with Steven Bergwijn and Lucas Moura partnering Harry Kane upfront, instead of Son Heung-Min and Dele Alli, who both dropped to the bench.
Bergwijn broke the deadlock for Antonio Conte's charges on the half-hour mark after finishing off a good combination play in the box but Jarrod Bowen levelled matters three minutes later.
Moura quickly restored the lead for Spurs a couple of minutes later after he was setup by Bergwijn and that was enough to book a spot in the semi-finals.
