The Reds kicked off their pre-season with a game against Chester, and were resounding winners over the National League North outfit on Saturday afternoon.
Karius played 45 minutes of the match as he was seen in a Liverpool shirt for the first time since making two costly errors in the Champions League final against Real Madrid at the end of May.
The German goalkeeper was widely criticised for his performance despite the possibility of a concussion, but he was back in action against Chester and made a couple of smart saves during his half of action.
"It was a difficult save but he made that and looked good, and the high ball was difficult. Many goalkeepers would have dropped it and then everyone would have been: 'Oh that happened' and it starts again," Klopp told reporters.
"We need to cool down the whole situation and play football and work on our weaknesses, use our strengths and everything will be good."
Liverpool will continue their preparations for the 2018-19 campaign with a clash against Tranmere Rovers on Tuesday night.