The visitors' one-goal advantage from last week's clash in Madrid had been reversed before the sound of the half-time whistle due to strikes from Julian Brandt and Ian Maatsen for Dortmund.
In a hectic turn of events at the beginning of the second period, Diego Simeone's men regained control of the tie courtesy of a Mats Hummels own goal and an effort from substitute Angel Correa.
Just as Atletico's experience in these high-pressure matches was appearing to come up trumps, Dortmund hit two of their own in a chaotic three-minute spell to win the last-eight clash, with Niclas Fullkrug and Marcel Sabitzer beating Jan Oblak in the visitors' net.
For the first time since 2012-13 - the season in which they reached the showpiece match at Wembley versus Bayern Munich - Dortmund will be involved at the semi-final stage of the Champions League, with Kylian Mbappe and Paris Saint-Germain standing in the way of a return to the English capital in June.
Following last week's first-leg defeat at the Wanda Metropolitano, Borussia Dortmund made the single change to their starting XI, with Brandt - who was influential during his side's second-half improvement in Spain - taking the place of midfielder Felix Nmecha as the 22-year-old dropped to the bench.
There were major absentees from Dortmund's matchday squad, though, with both last week's goalscorer Sebastian Haller and leading marksman Donyell Malen missing out on the second leg due to ankle and knee issues respectively, significantly reducing the Germans' firepower up top.
After netting his team's second of the reverse fixture in Madrid last Wednesday, wing-back Samuel Lino was unavailable for the trip to Signal Iduna Park due to suspension, meaning that the fit-again Mario Hermoso entered the XI, with veteran Cesar Azpilicueta moving onto the left of the visitors' defence.
Having netted a brilliant brace in Saturday's success over Borussia Monchengladbach in the Bundesliga, Sabitzer had a glaring chance to give Dortmund a second-leg lead after just three minutes following Karim Adeyemi's cutback from the left, however the former Manchester United man took a touch in the box, allowing Azpilicueta to recover and produce a goal-saving block.
Despite their clear defence-first approach in the opening exchanges, a two-man forward line of Alvaro Morata and Antoine Griezmann were linking up nicely on the break, with the former being sent through on goal in the fifth minute, however after moving across Sabitzer expertly, the Spaniard dinked his effort over Gregor Kobel but on the wrong side of the Dortmund post.
After piling on the pressure for the majority of the first half, Dortmund were ahead on the night and level on aggregate when Hummels played a lovely ball with the outside of his foot to Brandt in the Atletico box, and the German managed to get a shot off, with Oblak only able to parry the ball into the roof of the net.
Simeone's men were attempting to weather a serious onslaught from the Bundesliga side but found themselves two goals down in the second leg, with Chelsea loanee Maatsen finding the net for the first time in the Champions League, unleashing a low-driven strike into the bottom corner after being teed up by a Sabitzer backheel in the box.
Looking to swing the momentum in their favour during the second period, the Spanish visitors made three alterations at the half-time interval, with Morata, Azpilicueta and Nahuel Molina being replaced by Correa, Pablo Barrios and Rodrigo Riquelme, who slotted in at wing-back.
In truth, it was a touch of fortune which allowed Atletico to restore parity in the tie by netting their first of the evening, with Dortmund veteran Hummels scoring own goal on his 502th appearance for the German club after his laboured attempt to clear following a corner-kick unintentionally getting the better of Gregor Kobel.
Acting as a direct replacement for Morata, Argentina international Correa was having a real influence on proceedings at the beginning of the second half and got his name onto the scoresheet to give Simeone's troops the lead in the tie once again, volleying home a close-range effort following numerous desperate blocks in the Dortmund area.
With their wealth of Champions League experience, it felt as if the golden ticket for the semi-finals was falling at the feet of Atletico, however a remarkable three-minute period from the Bundesliga outfit completely switched the complexion of the quarter-final once again.
At the centre of everything positive for Dortmund once more, Brandt played Sabitzer down the left flank, and the Austrian's lofted cross was met by a perfectly-weighted header from Fullkrug, who got across Jose Gimenez to direct the ball past Oblak and into the net off the back post.
Unwilling to settle for an extra 30 minutes against their plucky visitors, Erdin Terzic's men pushed on for a goal that would secure their spot in the last four within the 90 minutes of normal time, and the passionate Signal Iduna Park crowd were treated to just that.
Finding himself in the box, Brandt controlled a long ball down to striker Fullkrug, who unleashed a fierce shot which was blocked, however the rebound fell to the onrushing Sabitzer, and the marauding midfielder shifted to the left before arrowing his strike into the bottom-right corner.
Before a semi-final clash in the Champions League with Ligue 1 giants PSG, Dortmund will refocus their attentions onto their pursuit of a top-four finish in the Bundesliga, with German champions Bayer Leverkusen the visitors to Signal Iduna Park on Sunday afternoon.
Also fighting to end up inside the top four of La Liga, Atletico will look to bounce back from a bitterly-disappointing night in Germany as they travel to Deportivo Alaves on the weekend.