In a remarkable finale, the 1997 European Champions booked their first last-four berth in 15 years courtesy of two goals in second half stoppage time to edge out Malaga 3-2 on aggregate after a goalless first leg in Spain.
While Santana's goal was well offside, their last-gasp victory was not only reminiscent of the best German comebacks but also fuelled by luck, coach Juergen Klopp said.
By the time they turned the tie on its head, hundreds of Borussia spectators had already left the stadium thinking their team was out of the competition.
"We have to accept that we could have been eliminated but we were lucky," Klopp said. "We played our worst Champions League game, that has to be clearly said.
"The team really wanted to advance and that led to us playing without our usual light touch."
The Germans, who are the only unbeaten team remaining in the competition this season, saw Malaga take a 25th minute lead but bounced back with a Robert Lewandowski strike before squandering a host of chances to score again.
The Spaniards, however, retook the lead in the 82nd minute with Eliseu's goal leaving a desperate Dortmund needing two more goals to qualify.
"We were never that easy to defend against like today, we lacked patience, creativity and flexibility," Klopp added.
"But the greatest games are not remembered for being fantastic because they were one-sided but because they were tight, and there was a turn of events in the end that no one expected. I will not forget this game."
It was indeed a game with more than one last-minute twist as Marco Reus levelled in the first minute of stoppage time before Santana's dramatic winner sent Dortmund through and triggered wild celebrations across the Ruhr valley city.
For Dortmund, who were on the brink of bankruptcy in 2005 before starting their financial and sporting ascent, it was a game that put them firmly back on the European map after dominating domestic competitions in the previous two seasons.
"We are not lucky that often," Klopp added. "When we were not good in the past we usually lost. This time luck was on our side."
- Sports & Recreation