It was 1-0 at the break after Mohamed Salah netted the opener eight minutes into the friendly, with Liverpool making four substitutions before heading back out for the second 45 minutes, including Darwin replacing Roberto Firmino up front.
The former Benfica striker did not have to wait long to get on the scoresheet, stepping up to the penalty spot and converting to make it 2-0 in the 48th minute.
Just three minutes later Darwin was the beneficiary after a turnover deep in the Leipzig defence, getting slipped through on goal by Trent Alexander-Arnold before finishing hard and low across the goalkeeper into the bottom-left corner.
He completed his hat-trick with a lunging tap-in from the edge of the six-yard box, getting on the end of a low cross from Harvey Elliott, before completing his rout with a fourth in stoppage time.
Speaking to the media after the win, Klopp said he was excited to see his side clicking in the second half with their near-£100million man, and that it is the best way to quell any chatter about his hefty purchase price.
"When we struggled a little bit, it was always after three minutes' break we played football again, then a deep breather and play football again," he said. "I wanted us to be a bit more hard with ourselves, go over that point – it's pre-season.
"I think we did that in the second half, obviously. 'Hendo' came on, and [James Milner] – they both were lively.
"Hendo all of a sudden as an eight, went in behind and we created chances. Then Mo gives Darwin the penalty and Darwin – Pandora's box was open. That'a of course a perfect night for him.
"We always think that if you pay a lot of money then the players feel no pressure or whatever – they are all completely normal human beings.
"If the first touch is not perfect then all of a sudden… this generation of players read social media, which is really not smart, but they do. All of a sudden you get in a rush and these kinds of things.
"That's the best way obviously to stop all these discussions. He's a different striker to what we have or what we had, but he's a really good one."
Liverpool's next friendly will take place away against Salzburg next Thursday AEST.