Karius started and played 45 minutes in a 7-0 friendly win over Chester on Saturday as he began the road to redemption following his nightmare display against Real Madrid in May.
The German came under fire for committing two costly errors in that 3-1 defeat, although medical experts later determined the effects of a possible concussion may have impacted his performance.
But he took his place in the starting line-up at Swansway Chester Stadium and encountered no vitriol from the travelling Reds fans, making an early one-on-one save and completing his aerial duties with minimal fuss.
"I am not sure I should have expected something different but it was nice," Klopp said of the supporters' welcome for Karius.
"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.
"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's next pre-season assignment sees them visit Tranmere Rovers on Tuesday.