By FOX Sports Staff
The 23-year-old found the back of the net three times for Lazio in a 3-1 win over the Bundesliga outfit to cap a 5-1 aggregate triumph.
Kozak got the Italians off the mark after just six minutes when he managed to get onto Stefan Radu low cross before doubling his team's advantage with a fierce finish from 15-yards out.
Tamas Hajnal pulled one back for Stuttgart just after the hour mark, but Kozak got his third goal in the 87th minute with a header from Ogenyi Onazi's cross.
"This is my first hat-trick for Lazio and of course I am really happy," Kozak was quoted as saying on the Europa website.
"I am going to enjoy tonight [Thursday], I am delighted for myself and my team that we have gone through.
He added: "It is not like I give less in the league, I always try to give my best, even if something seems to be lacking in Serie A. Every minute I play, I try to do the same thing."
"We are now into the quarter-finals and all the teams are strong, so it is going to be a great challenge for Lazio, for us and for the fans, and we will give it our best no matter who we face. I think everybody has seen how good we are and that we can make it all the way."
Thursday's match was played in an empty Stadio Olimpico in Rome. Last month, UEFA 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.
Lazio will find out who they next play in the competition on Friday.