An entertaining game saw Monchengladbach awarded three penalties, with Lazio acquiring a red card and two goals in the last few minutes.
Both sides were restricted to long-rang efforts after that, but a good move down the left did find Luuk de Jong free inside the box. Unfortunately the Dutchman headed wide.
Lazio ended the half on top, but couldn't find the equaliser with Sergio Floccari missing a couple of chances.
Shortly after the break Floccari made up for his misses, firing the ball home from inside the box to draw Lazio level.
Just five minutes later, they were ahead, in almost exactly the same fashion. A cross into the box found Libor Kozak this time, who found the back of the net with a powerful shot.
With 20 minutes to go, the home side were handed a way back into the match when Andre Dias was shown a second yellow card, after conceding a penalty. This time though, Federico Marchetti was equal to Stranzl's spot kick.
The German side pushed home their man advantage and were eventually awarded their third penalty of the night, when Lorik Cana brought down De Jong.
Substitute Thorben Marx claimed penalty duty after Stranzl's miss, and duly equalised for the home team.
Monchengladbach weren't done there, and with just a few minutes remaining on the clock, they took the lead thanks to a Juan Arango free kick.
The drama continued still when, in the 94th minute, Kozak won a free kick before getting into the box and heading Hernanes's cross into the net, with what was the last touch of the game.