Dortmund were 2-1 down in added time on Tuesday when Marco Reus and then
Felipe Santana grabbed 91st and 93rd-minute goals respectively to put them through 3-2 on aggregate after last Wednesday's goalless first leg.
Replays showed both Dortmund's dramatic late goals were offside, leaving Malaga coach Manuel Pellegrini calling for European football's governing body UEFA to investigate, but Malaga's second by Eliseu was also shown to have been offside.
"Football showed both sides of its face, the bitterly disappointed defeated and the overjoyed winners," said Klopp, as his side reached the last four of the Champions League for the first time since 1998.
"It's not often we have needed so much luck, we weren't so good, but despite that, we won. Over the two games I think we are a deserving winner, but if we play like tonight again we won't win the Champions League," he said.
"Now we have reached our target of a semi-final. It's crazy. I think I need to see a doctor!"
Malaga coach Manuel Pelligrini, who had a round-trip flight to Chile in the days before the match after the death of his father, was highly critical of Scottish referee Craig Thomson and his assistants.
"It was like there wasn't a referee on the pitch at the end, it was chaos in the closing stages, there were six or seven things which went unpunished in our area," he fumed. "There was a chain of mistakes. Whatever needs to be changed, must be changed."
But Klopp gave some perspective to Pellegrini's comments as the Spaniards were denied a place in the last four on their debut season in the Champions League.
"I can understand their disappointment, but I didn't think the referee had a bad game," said Klopp. "I heard our first goal (in added time) might have been offside, I didn't see it. I can understand when you lose a game like that, you feel bitterly hard done."
Having stabbed the ball over the line for the dramatic winner, Brazilian centre-back Santana was quickly buried under a mountain of Dortmund bodies as his team rushed to celebrate after the final whistle.
"Up until now, our story has been like a Hollywood movie - hopefully it will have a Hollywood ending," said Santana's central defensive partner Neven Subotic.
"I just can't really believe it. This is the craziest moment in all of my footballing career. It has been a special match for everybody - for us and the fans."
Malaga winger Joaquin had given the Spaniards an early away goal before Poland striker Robert Lewandowski's equaliser made it 1-1 at the break.
Second-half substitute Eliseu then looked to have put Malaga in the semis with a goal eight minutes from time, before Dortmund produced two goals in two magical minutes.
"We were four minutes away from the semis," said Malaga striker Roque Santa Cruz. "There is huge disappointment in the dressing room. We had the tie in our hands."