The hosts were stunned when Kasey McAteer fired the Foxes in front in only the third minute.
But sustained pressure from Jurgen Klopp’s side eventually told as Cody Gakpo levelled before Szoboszlai came off the bench to put them in front with an unstoppable shot in the 70th minute.
Diogo Jota added a third in the 89th minute and it was no less than Liverpool, winners of this competition a record nine times, deserved after they poured forward in response to the early setback, having 27 attempts at goal in all.
McAteer’s early goal remained Leicester’s only shot on target by the final whistle.
Jurgen Klopp had made 10 changes to the side that beat West Ham 3-1 at the weekend to maintain their impressive start to the Premier League campaign but Enzo Maresca, whose side are top of the Championship as they eye an instant return to the top flight, matched him with just as many.
And the in-form visitors silenced the windswept Anfield crowd as a Liverpool free-kick turned into a Leicester goal.
Kostas Tsimikas’ delivery was punched clear and the Greece defender was left in a heap by Marc Albrighton as Yunus Akgun raced clear before slipping the perfect ball into the path of 21-year-old academy product McAteer, who had time to pick a spot for his fifth goal of the season.
A Liverpool response was guaranteed, but Wataru Endo, making his third start since joining from Stuttgart, fired a shot narrowly wide before Harry Souttar blocked Gakpo’s shot after neat passing cut open Leicester’s defence.
The following corner was worked short to find Jota at the far post but the Portuguese forward fell over the ball before Jakub Stolarczyk blocked Ben Doak’s shot as the 17-year-old picked up the pieces.
Doak then went even closer from the next corner, hitting the crossbar on the rebound as Stolarczyk could only parry a shot from Liverpool’s 20-year-old defender Jarell Quansah.
Gakpo thought he had equalised in the 22nd minute as he headed Tsimikas’ free-kick goalwards but Conor Coady – belatedly making his Leicester debut at his boyhood club following injury – scrambled it off the line.
A mistake from Ricardo Pereira led to Liverpool’s next opportunity as helost control inside the area and Harvey Elliott played in Gakpo, but his shot was deflected over.
Liverpool trailed at the break despite having 15 shots to Leicester’s two, but needed only three minutes of the second half to level.
Ryan Gravenberch, making his first Anfield start, fizzed in a pass to Gakpo, who stuck out a leg to control with his back to goal before twisting to find the bottom corner of the net.
Liverpool were firmly on top now and Gakpo should have had a second just before the hour, getting a glancing header on Elliott’s cross but watching it hit the underside of the crossbar and bounce down on to the line before Leicester cleared.
Both managers turned to their benches to strengthen, with Maresca sending on Kiernan Dewsbury-Hall and Patson Daka, while Klopp called on Szoboszlai and Darwin Nunez.
Klopp was the manager to see his moves pay off as within five minutes of coming on, Szoboszlai unleashed an unstoppable shot into the top left-hand corner from the edge of the D.
Liverpool continued to pile on the pressure, and Jota sealed it in the 89th minute as he flicked Quansah’s low cross in off the inside of the post.