Armed with a 2-0 lead from the first leg in Germany, Lazio made an ideal start and left Stuttgart's hopes in tatters before they even had a chance to settle.
Tamas Hajnal pulled one back for Stuttgart just after the hour mark, but they needed five. Kozak's third killed off the hopes of even the most optimistic Swabian.
The banks of empty seats at the Stadio Olimpico in Rome told the story of Lazio's campaign off the field of play.
UEFA last month ordered the Italians to lock out supporters from their next two home matches in Europe after crowd trouble during their last-32 match against Borussia Monchengladbach.
The Serie A club had been charged over the setting-off of fireworks, racist behaviour and insufficient organisation following the second leg in Rome.
Those responsible, and those unfortunate by association to be denied the chance to see their team, missed a treat in the early stages.
Stuttgart threatened to move ahead in the fourth minute when Federico Marchetti had to tip a 25-yard blast from Raphael Holzhauser behind for a corner.
But within 90 seconds Stuttgart were trailing, after Stefan Radu drilled a low cross from near the left corner flag that Kozak barged through to meet and blast first time beyond Sven Ulreich.
From eight yards, Kozak thrashed the ball high into the centre of the goal.
One crushing blow to Stuttgart's slim hopes was followed by effectively the fatal strike two minutes later as Hernanes fed a superb cross-field ball into the feet of Kozak who, from a position to the right of goal, lashed a 15-yard shot in off the underside of the crossbar.
The battle was won.
Holzhauser and Shinji Okazaki peppered shots at the Lazio goal but the most notable action in the remainder of the first half saw home goalkeeper Marchetti substituted after a particularly uncomfortable collision with Stuttgart forward Vedad Ibisevic. Albano Bizzarri took his place.
Hajnal scored for Stuttgart in the 62nd minute, a low left-footed drive to the bottom right corner from 15 yards, and the German club's players optimistically raced to retrieve the ball from the back of the net.
Bizzarri saved twice from Christian Gentner, but they were playing out the formalities of an impressive two-leg success, capped by Kozak's third in the 87th minute, a header from Ogenyi Onazi's cross.