The visitors extended their advantage after the break through Ogenyi Onazi, who finished cooly after beating three players with a mazy run.
Look back at an impressive away performance from Vladimir Petkovic's side with Sports Mole's minute-by-minute commentary below.
"We do not think that we have much of a chance. This is a very big game for us, which hopefully will drive us forward. We want to show that we can compete against a very good opponent."
He said: "My team must be aware of its abilities and continue the positive streak in Europe. This game will not be decisive, but it will give us some important answers ahead of the second leg.
"Lazio have proved themselves many times. Even in the 3-0 defeat to Milan, I wasn’t disappointed with the way we played while we had 11 men on the field. We must fight to score a goal and not concede, as this tie is played over 180 minutes.
"I am not worried by our results away from home, but we will become a big club when we are able to perform the same way in any stadium."
STUTTGART: Sakai, Tasci, Rüdiger, Boka, Kvist, Gentner, Okazaki, Maxim, Traoré, Macheda
"But I’m a professional, I’m wearing the Stuttgart jersey tomorrow and I will give everything."