Carlos Tevez put the Italian side ahead early on before a defensive mix-up gifted Dortmund an away goal through Marco Reus in the 18th minute.
Alvaro Morata scored against the run of play two minutes before half-time to regain Juve’s lead.
Dortmund have only themselves to blame for leaving the Italian city with a defeat with both of Juve’s goals coming far too easily, according to Klopp.
“We did a lot of things right, especially in the first half, but the two goals were easy, too easy,” Klopp said. “Our defending was not perfect. We made it easy for Juve.”
Klopp was forced to change his game plan in the second half after making two substitutions, with defenders Lukasz Piszczek and Sokratis Papastathopoulos going off injured.
Central defender Neven Subotic and forward Kevin Kampl, both out with a stomach bug, will be back for the second leg as will Papastathopoulos, whose muscle injury is seen as not too serious.
“For one good half and a mediocre second half it is a result we can live with. We are still in the running,” said Klopp.
“We made it a tight race with the goal we scored and I don’t think we are returning home unhappy.”