Libor Kozak scored a hat-trick as Lazio turned on the style behind closed doors to win 3-1 on the night and 5-1 on aggregate to reach the Europa League quarter-finals.
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.
Benfica eased into the last eight as they proved too strong for Bordeaux in the south of France. With the comfort of a 1-0 lead from the first leg, the Portuguese withstood Bordeaux's early attacks before snatching a lead through Jardel's header.
Cheick Diabate gave the hosts a slither of hope with 16 minutes left but it was extinguished seconds later by Oscar Cardoza as Benfica made certain of the win. The teams exchanged further goals in the dying seconds, with Jardel scoring an own goal before Cardoza replied instantly with Benfica's third.
Two extra-time goals by Jose Rondon and Vladimir Dyadyun steered Rubin Kazan through with victory over Levante in Moscow.
Rondon struck in the 99th minute for his fifth goal in the Europa League this season with substitute Dyadyun sealing victory with eight of the 30 extra minutes remaining after Levante had committed men to finding the equaliser which would have seen them through on away goals.
Ten-man Basle produced a desperate defensive display to lose 1-0 but book their place in the quarter-finals 2-1 on aggregate at the expense of a wasteful Zenit St Petersburg.
Basle goalkeeper Yann Sommer saved a late penalty from Roman Shirokov and the hosts hit a post and missed an open goal as they threw all they had at their opponents, but a tie-levelling second goal eluded them.
Trailing 2-0 from the first leg, Zenit halved the deficit after half an hour through Axel Witsel and things looked to be going their way when Basle had Marcelo Diaz sent off for two yellow cards just before half-time. But the Swiss champions defended superbly after the break, limiting Zenit to just a handful of clear-cut chances to progress.
Substitute Salih Ucan was the hero as Fenerbahce did enough to squeeze through with a 1-1 home draw against Viktoria Plzen.