Kostas Tsimikas scored the winning spot-kick after a tense match in which the two sides went close, playing almost a carbon copy of the Carabao Cup final the two played in February.
Liverpool, who still have a Champions League final to play, saw Mohamed Salah, Virgil van Dijk and Andy Robertson sent off with injuries.
The win completes the set for Klopp, who has now won every major trophy at his disposal since taking over at Anfield in 2015.
In doing so, the German became only the second manager to win the Champions League, Premier League, League Cup and FA Cup with one team, after Sir Alex Ferguson.
Speaking to the BBC after the game, Kopp said: “Exceptional, it was an incredible and intense game against Chelsea – they would have deserved it [the win] exactly the same way, as in the Carabao Cup – that’s how the margins are small.
“I couldn’t be more proud of my boys, the change they’ve brought about, the strength they’ve fought with, the early changes. I think Virgil is fine but his muscle has been injured.
“All of those things, missing good chances, overcoming good times from Chelsea and then having great times ourselves.
“Then in the penalty shootout it was nerve-wracking, my nails are gone but I really feel for Chelsea – for the second time, 120 minutes and you get nothing, it’s too hard. But for us, I am quite happy.
“We are mentality monsters but there were also mentality monsters in blue – it was a penalty. Chelsea played exceptionally but in the end there has to be a winner and that was us today. “