Libor Kozak's double-goal salvo put the tie out of sight within just eight first-half minutes at the Olimpico, and despite Tamas Hajnal's consolation effort midway through the second period, the Czech struck again late on to ensure a comprehensive 5-1 aggregate triumph.
The opener came just six minutes into the action as Stefan Radu burst down the left flank and fizzed in a cross which Kozak met at the back post with great aplomb to put the Biancocelesti on their way.
Kozak did not waste any time in grabbing his second of the evening as Lazio proceeded to cruelly hit Stuttgart on the break, which ended in the hitman coolly dispatching the finish past Sven Ulreich.
A string of snapshots ensued as the visitors looked for a way back into the tie but to no avail, with Raphael Holzhauser and Hungarian midfielder Hajnal firing the best of die Roten's attempts wide of the target.
But on the hour mark, Stuttgart was presented a route back into the match when Hajnal took control of the ball on the edge of the box and slotted home to fractionally decrease his side's deficit.
Lazio, was happy to take its foot off the gas and adopt a counterattacking game while Stuttgart desperately stepped up their efforts, and came close through Antonio Candreva on multiple occasions.
Nonetheless, Bruno Labbadia's charges showed intent to salvage yet more pride with the minutes ticking away, and both Shinji Okazaki and Vedad Ibisevic could count themselves unlucky as Federico Marchetti stayed on his toes.
Unfortunately for the Germans, Kozak had not quite finished his work for the evening when he found the back of net with just three minutes of time remaining for his 10th European strike in just his ninth continental appearance.