Despite enjoying most of the possession and creating several openings, Bruno Labbadia's men conceded twice to what the coach called an "effective" team, and the Reds are now on the verge of leaving the European stage.
But when there is a will, there is a way, and Stuttgart will travel to Rome next Thursday still believing they can turn things around.
"We paid the price tonight [Thursday]," Labbadia said. "You could see in this game what difference cleverness can make.
"But we will try everything in the second leg, even though we have to accept it is going to be hard."
Stuttgart's best results in the Europa League this season have all come on the road, and a 5-1 win at Steaua Bucharest in a defining group stage victory will fuel their hopes of securing victory in Italy.
Nine of their 12 goals in the competition have come on their travels, meaning sightseeing will be the last item on their agenda next week.
"We're not just going there to see the Colosseum and an empty stadium," the club's director of sport Fredi Bobic added.
"We need to play a perfect game."