Kagawa provided the assist for Ciro Immobile's opener at the Constant Vanden Stock Stadium, while Adrian Ramos bagged a brace to seal the victory.
And Klopp was impressed by Kagawa's performance as the Japan international continues to improve after making his return from an ill-fated spell at Manchester United.
"I thought Kagawa did very well," Klopp told reporters after the game.
"He played some great passes.
"He produced a big performance. He was very important for us, without doubt."
Klopp also singled out veteran midfielder Sebastian Kehl for praise, but has some concern over the 34-year-old, who picked up a knock on the night.
"Kehl had a very good game, but I only want to continue talking about him when I know that he can play in the next game," he added.
"I am a very superstitious man. He got a blow on the foot, so I will wait and see."