The Reds found themselves behind inside the first ten minutes with James Maddison sliding Jamie Vardy into the box before the Foxes striker drilled a low finish past Caoimhin Kelleher's outstretched hand to make it 1-0 to the visitors.
Leicester doubled their advantage shortly after as they put themselves in full control of proceedings, with Vardy clinching his second of the game after getting on the end of Daka's squared pass to make it 2-0 from close range.
But it didn't take long for Liverpool to pull one back and drag themselves back into the game. Roberto Firmino turned away from goal when it looked like a first-time finish from Conor Bradley's knockdown may be on, with the Brazilian picking out Alex Oxlade-Chamberlain, who turned the ball home with an excellent low finish from the edge of the box.
Liverpool were unable to formulate any real openings and they fell further behind through James Maddison, who picked up the ball at the edge of the box and sent a powerful strike into the back of the net to make it 3-1 to Brendan Rodgers' side.
Ibrahima Konate, James Milner and Diogo Jota came on at the break for the Reds and their introduction undoubtedly had a positive impact for Jurgen Klopp, with the Reds beginning to build momentum despite the final finish often lacking.
It was Jota that ultimately halved the deficit for Liverpool as Takumi Minamino played the Portuguese through on goal, with the Reds forward calmly receiving the ball and slotting his finish into the back of the net to make it 3-2 with a little over 20 minutes left on the clock.
With less than two minutes left of stoppage time, James Milner's long ball was chested down by Minamino inside the box before the Japan international slotted the ball into the back of the net with an excellent finish to force the game to penalties.
Kelleher denied Luke Thomas from the spot before Minamino failed to convert from the Reds' final penalty as the shootout went to sudden death, but Kelleher saved again from Ryan Bertrand as Jota stepped up and sent Liverpool into the semi-finals, where they will meet Arsenal.