The enthralling Wembley showpiece saw four goals disallowed across a goalless 120 minutes before 21 consecutive penalties were scored in an incredible penalty shootout.
Kelleher, who started the game ahead of Alisson as Jurgen Klopp kept faith with the man who had helped the Reds to Wembley, converted the 21st spot-kick.
Kepa Arrizabalaga was brought on to replace Edouard Mendy specifically for the shootout deep into extra-time but shot over to hand Liverpool its first domestic cup in 10 years.
Kelleher said in the aftermath that he was delighted with the success, but admitted he had not realised that he was the matchwinner at the time.
"I thought I'd saved one, I got close to a few and then when it came down to me, I didn't even realise I'd scored the winning penalty!", he said.
"I forgot that I'd scored the winning one, all the penalties from the lads were class, and I was just happy to score."
When asked if he had channelled his youth as an outfield player when taking his kick, the 23-year-old responded: "I think it was more hit and hope.
"I got a hand on a few, but all the penalties were very high quality, and I'm just thankful we were able to win.
"I thought we had scored when we got that goal [Joel Matip's second-half disallowed effort], and it was obviously disallowed. They had a few disallowed too, so I think over the game a draw was a fair result."
Kelleher also revealed his exchange with Klopp after the tense finale, with the Liverpool manager telling his young keeper that he had written his name into the club's history.
"He [Klopp] just said 'well done for scoring the winning penalty, there's a wall at Liverpool with all the goalkeepers who have managed to win cups, and he said 'now's your chance to join them!'," he added.
The 11-10 shootout win over Chelsea represented the highest-scoring penalty shootout in history between two English top-flight teams, with Liverpool's back-up keeper eventually proving the unlikely hero in the Reds' record ninth League Cup win.