The Bundesliga outfit fell behind to a Carlos Tevez effort after 13 minutes - the Argentine tapping home after goalkeeper Roman Weidenfeller had pushed the ball into his path.
Marco Reus pounced on a Giorgio Chiellini slip to restore parity five minutes later, but Alvaro Morata diverted in from a Paul Pogba cross two minutes from the break to give Juve a narrow advantage heading into the second leg.
"There were a few situations in the first half when we could have delivered the attack better," said Klopp.
"We did well in the first half and caused Juve problems. But the goals we conceded are tough to swallow.
"We showed a good reaction to going 1-0 down and played well. Ahead of their second goal we were too passive. We can and have to defend better in that situation.
"In the second half we only had one or two opportunities, while Juve had two really good chances. From that perspective, the result is okay."
And Reus echoed his coach's comments, stressing Dortmund need to shore up their defence for the return fixture in three weeks' time.
"We know we can defend better than we did ahead of the two goals we conceded," he added. "Now we have to fix this in the second leg.
"Our goal didn't give us as much confidence as we thought it would. In the second half we were not dangerous enough, but everything is still up for grabs in this tie."
