A tight encounter saw several big chances missed and four goals disallowed before Kepa, who was brought on specifically for the shootout, missed the decisive penalty, which Liverpool won 11-10 after a goalless 120 minutes.
The remarkable miss handed Liverpool a record ninth League Cup competitions – one more than Manchester City – with this Jurgen Klopp's first domestic cup since arriving at Anfield in 2015.
European and world champion Chelsea will rue its wasted opportunities even before the spot-kicks in its fourth consecutive domestic final defeat.
The first of them came inside the opening 10 minutes when Christian Pulisic was denied at point-blank range by Caoimhin Kelleher in Liverpool's goal, and the chance would set the tone for an extraordinary match.
Chelsea's starting goalkeeper Edouard Mendy went one better than his counterpart on the half-hour mark, diving full length to keep out a long-range drive from Nady Keita before somehow getting up to block Sadio Mane's follow-up strike and completing an impossible double save.
Kai Havertz missed another golden opportunity for the Blues just before the break and somehow the teams went in goalless at half-time.
Chelsea had the first big opportunity of the second period – and arguably of the match – when Mason Mount was found unmarked in the box by Pulisic, and while he beat Kelleher, his shot struck the inside of the post and bounced away to safety.
Then came a flurry of disallowed goals the first going against Liverpool after Joel Matip had nodded the ball home from close range, but it was ruled out for an offside as Trent Alexander-Arnold took the free-kick that led to the chance.
Romelu Lukaku, who came off the bench, was then ruled offside by a hair's breadth after he'd blasted the ball into the back of the net in extra-time, and Havertz also scored before seeing his effort chalked off when the flag went up for a much more clear offside.
The teams couldn't be separated after extra-time, and after Arrizabalaga was brought on for Mendy, a penalty shootout also couldn't pull them apart, with 20 spot-kicks all scored in the shootout, leaving the goalkeepers to duel it out.
Kelleher was first, and easily beat Arrizabalaga, leaving it up to the Spaniard to try his luck. Unfortunately for him, he blazed over the bar to send Liverpool and its fans into delirium after the most extraordinary cup final in recent memory.