With star goalkeeper Alisson injured, Adrian was making his first start for Liverpool and the Spaniard stepped up – much to the delight of Klopp.
"Adriaaaan! Like Rocky," the Liverpool manager told BT Sport, referencing the famous boxing films starring Sylvester Stallone. "What a story. He kept his nerves together.
"He's loud in the dressing room already. I don't think he's won a lot in his life so it's good for him to win it.
"He made some unbelievable saves, both keepers did, so I'm really happy for him."
Liverpool fell behind to an Olivier Giroud goal before Sadio Mane's brace had them ahead in extra time.
However, Jorginho's 101st-minute spot-kick brought Chelsea level, leading to the penalty shoot-out.
"It was a very difficult game for both teams. It was all about winning it and we did that in the end," Klopp said.
"Nobody in the stadium wanted extra time, it was a killer. I'm not sure about the penalty but who cares now.
"We started well then dropped off. We could talk about football but it's too late now. We had to fight and the boys did."