The Welsh forward struck nine minutes from time in his side's quarter-final victory on Tuesday to clinch a 2-0 win, while also becoming the Reds' youngest ever goalscorer in the process.
At 17 years and 45 days, Woodburn's strike has smashed the club record, previously held by Michael Owen, by 98 days.
Speaking after the game, Klopp told reporters: "We all know how young he is, but it doesn't look like this in training sessions. It's a nice story and all the boys are really happy for him. everybody is smiling in the dressing room."
Lucas also had kind words for Woodburn, adding: "He is a brilliant young player. He did very well. He just needs to keep learning, keep improving. A goal will give him confidence to keep learning, to keep going."
In seeing off Leeds, the Reds booked their place in the semi-finals of the League Cup for a record 17th time.