Saul Niguez struck after just four minutes at the raucous Wanda Metropolitano Stadium and after that Atletico retreated into an effective rearguard action.
Reds boss Klopp had no issue with the defensive tactics but became increasingly irritated throughout the game that referee Szymon Marciniak did not clamp down on time-wasting and other histrionics.
He revealed his thoughts when asked why he had taken off Sadio Mane, who had been booked, at half-time.
Klopp said: "Obviously the plan was to get Sadio out of the game with a yellow card. I was afraid his opponent would go down even if Sadio takes a deep breath or whatever, so I didn't want to have that situation. That's why we took him off.
"It was difficult for the referee to deal with the atmosphere tonight. After 30 minutes three Atletico players were already on the ground without being injured. The ref just needs to feel that game a little bit more."
Klopp called on Liverpool fans to show the same fervour as the Atletico supporters when the second leg comes around in three weeks' time at Anfield.
He said: "I had no problem with the result. I saw so many happy faces from Atletico tonight, I get that because it's a big win, but it's not over yet. That's the only thing I feel.
"The crowd wanted to help their team tonight. That makes it a very emotional game.
"Emotions are important. Tonight they were obviously completely on the side of Atletico but I am really looking forward to the second leg.
"We speak from time to time about the power of Anfield and the power a stadium can have, tonight you saw that.
"It's half-time and we are 1-0 down. Why should we give up? Even better the second half will be in a different stadium, our stadium, that will be different and Atletico will feel that and that's what we are looking forward to."
Liverpool will assess Jordan Henderson after the captain left the field with a hamstring injury 10 minutes from time.
Klopp said: "Hendo felt his hamstring unfortunately. We have to wait. He was not too concerned but enough to go off obviously."
The win was a crucial one for Atletico boss Diego Simeone, who has been coming under pressure due to his side's indifferent form in recent months.
The Argentinian paid tribute to the attitude of his battling players and the fans for creating a ferocious atmosphere.
Simeone said: "My side started winning when we came up the road leading to the stadium.
"We saw players who believed. We saw the beautiful side of football. The crowd and the side were a block as one.
"Winning breeds winning. The best way to feel strong is to win and that's what we want to do. It was a very important win for the club tonight.
"In the eight years I've been here, I've not seen the crowd like that for the whole game. It was really emotional.
"It's not the best night ever because we didn't win a trophy but there are nights you don't forget.
"The best side in the world came here, and we beat them. But it's only one game down, one to go. Liverpool had their chances. They were dangerous, they've got good players all over the pitch."