If Barcelona had scored a goal of similar fashion then the world would have stopped and applauded -- Dortmund are not there yet, but this team is showing a potential which should excite those who embrace such pure football.
But while its attacking prowess was clear to see, the backdoor always appeared ajar for a Malaga side determined to make light of its financial troubles and secure a last four berth.
Joaquin twice failed to scored with headers when well placed, but those misses appeared not to matter when Eliseu touched home Julio Baptista's shot with eight minutes remaining.
That goal, which was converted from an offside position, left the home supporters crushed.
But Dortmund refused to be denied and with the game drifting into the 91st minute, Reus lashed home from close range to level the game at 2-2 and leave his side needing one more goal.
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.