Klopp, whose team are now bottom of Group F with little chance of qualifying, hailed Van Persie.
"He is one of the best strikers in Europe. He is a great player," the Dortmund boss said.
"I have hardly seen a player who plays so deep in midfield and is still always dangerous in the box. He is simply unbelievable.
"Once the cross was in the air for the first goal it was inevitable that he was going to score. That really hurt us and gave them confidence."