Kelleher, 23, and his opposite number Edouard Mendy made a string of saves over 120 minutes as the sides played out a pulsating 0-0 draw and had to be separated by penalties.
The Irishman scored the Reds 11th consecutive spot-kick before Chelsea goalkeeper Kepa, who had replaced Mendy in the dying moments of extra-time, blazed his effort over the bar.
“I don’t know when I saw him the first time but it’s quite a journey from the boy I saw the first time to the man he is today,” said Klopp, who kept the faith in cup ‘keeper Kelleher ahead of No1 Alisson Becker.
“Alisson Becker is the best goalkeeper in the world for me. There are a lot of good goalies but this goalie is absolutely insane. And Kelleher is the best No2 in the world for me as well, especially for the way we play.
“He had an incredible game.
“People asked me two weeks ago if he was going to get another game to get some rhythm and that’s actually the first time I thought, ‘Oh he doesn’t have rhythm’.
“That’s the life of a young No2 to be ready when called. The game he played tonight was absolutely incredible. I can remember two incredible saves.
“He proved that the decision to line him up was absolutely right and then in a spectacular penalty shootout he showed the whole range of his skills.
“He finished one off with his feet which was top class.”
The Carabao Cup is Liverpool’s first domestic cup trophy under Klopp and means they are now the most successful club in the competition, having won it nine times.
Klopp hailed his squad and the experience of his side to see off the European champions.
“Facing Chelsea was like two lions going for each other, it was crazy,” he said. “They started better, we took over, then the second half with two good moments of Chelsea. At the end, everyone was tired and you have to get through it. The penalty shootout was one of the most speculator I ever saw.
“We’re more experienced now. We don’t get nervous when things don’t go well. The start of the game was clearly better from Chelsea but the first half was 63 [percent] possession or something.
“That’s not easy against Chelsea. Over 90 minutes you can’t hold them back away from your goal. It was pretty harsh to take from them I imagine.
“It’s a big one for us, first time for this group but the ninth time for the club which is very important as well. Our fans were obviously quite happy, the atmosphere was outstanding, I loved it and after the game its as nice to celebrate with the people who haven’t had the opportunity to celebrate for so long.”