Given Lazio's imposing home form, Stuttgart knew they had to make gains on home turf but barely created a clear opening in slumping to defeat.
Goals in either half from Honorato Ederson and Ogenyi Onazi sealed the win for Lazio.
Fenerbahce are another team dreaming of the last eight after a 1-0 win at Viktoria Plzen.
The Czech hosts failed to hit their stride at any point in the contest and Pierre Webo's late strike for the Turkish visitors was fully deserved.
Basle's strong home form in Europe saw the Swiss side to a 2-0 win over Russian champions Zenit St Petersburg.
Marcelo Diaz scored in the 82nd minute after both sides had wasted a string of chances, and substitute Alexander Frei netted from the penalty spot in stoppage time after Luis Neto had seen red for a foul on Mohamed Salah.
Benfica defeated Bordeaux 1-0, an unfortunate own goal from goalkeeper Cedric Carasso proving the difference between the sides at the Estadio da Luz
The day's early kick-off in Moscow between Anzhi Makhachkala and Newcastle ended goalless.
Levante and Rubin Kazan also played out a 0-0 draw in a match that saw both teams finish with 10 men.
The Spanish side had a chance to take control ahead of the second leg when Rubin's Cristian Ansaldi was given a second yellow card in the 53rd minute but they threw away the advantage almost immediately as Michel received a straight red card moments later.