With just a couple of minutes left to play, Malaga needed only to hold out and preserve the 2-2 scoreline to progress.
But as the yellow and black shirts poured forward, the Malaga defense buckled.
From a seemingly hopeless position, the ball was lumped into the penalty area before falling at the feet of Santana just a yard from goal.
He was offside -- earlier in the move there were four Dortmund players offside -- but that didn't stop him.
The Brazilian smashed the ball home, the assistant referee's flag didn't go up, the referee didn't blow. The goal stood as Malaga fell. Dortmund had triumphed.
"There is one very disappointed team in this moment and one lucky team," Klopp told Sky Sports.
"It's unbelievable. It was our worst game in the Champions League this season and not our best football.
"If we play like tonight we won't win the Champions League, but we have reached our target of the semifinal."
Malaga, which was making its debut in the competition, was left shell shocked as it suffered a heartbreaking exit.
"We were four minutes away from the semis," striker Roque Santa Cruz told Sky Sports.
"There is huge disappointment in the dressing room. We had the tie in our hands."
In the night's other game, Real Madrid progressed to the final four, despite suffering a 3-2 defeat in Turkey against Galatasaray.
Leading 3-0 from the first leg, Cristiano Ronaldo's early strike appeared to kill off any hope the home side might have had of causing an upset.
But Fatih Terim's men produced an inspired performance with goals from Emmanuel Eboue, Wesley Sneijder and Didier Drogba firing it into a 3-1 lead on the night.
Ronaldo grabbed a late second, his 48th goal in 47 games this season, to finally end the fightback.
Real finished with 10-men after Alvaro Arbeloa was sent off.
"I wasn't getting nervous because I have no time for it," Real boss Jose Mourinho told Sky Sports.
"I am on the bench so I have no time to think or to feel it, they are strong, strong mentally.
"They don't play against 11, they play 50,000 against 11 and it was hard for us.
"We are there, in the semifinals and that is what that matters.
"Today is the kind of match where you have everything to lose and nothing to win.
"People think we are already in the semifinal and players that that feeling from the outside. I felt the team was ready to work hard and had the motivation to reach Wembley in the final.
"I don't know who we will play in the semifinal, it will be against a great team, that's for sure and we need to play the semifinal and enjoy it."