Stuttgart knew they needed a result at home to have much hope, given the Italians' imposing home form, 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, who will now take a commanding advantage back to the Stadio Olimipico where they are unbeaten in 10 European outings.
They were slow to start, with Stuttgart enjoying early chances for Ibrahima Traore and Shinji Okazaki, the latter forcing a fine save from Federico Marchetti with a great header.
But Lazio scored with their first clear attack, with Ederson pouncing after William Kvist and Arthur Boka missed chances to clear the ball, turning his man and lashing a left-footed shot into the net.
Antonio Rudiger then blazed over for Stuttgart before Lorik Cana failed to keep a header down at the other end.
But Lazio were growing in confidence and created further chances only to see Anderson Hernanes and Ederson again miss the target.
Stuttgart briefly threatened before the break but Okazaki could not get his header on target.
Senad Lulic ought to have doubled Lazio's lead five minutes into the second half when he was handed a great chance, but he turned his shot wide after being picked out close to goal by Hernanes' free-kick following a foul on Antonio Candreva.
As it was, the second goal would come in the 56th minute, with Onazi doing pretty much all the work himself, the youngster picking up the ball in his own half and running clear to beat Sven Ulreich in the Stuttgart goal.
Okazaki wasted another chance for Stuttgart before Lazio put the pressure back on, Candreva missing the target before Tamas Hajnal's drilled free-kick struck the wall.
Libor Kozak was unlucky not to add a third when Ulreich made a fine save to keep out his shot before Hajnal missed the chance to halve the deficit by sending a late effort over the crossbar.