Chelsea, Benfica and Fenerbahce all won by two goals at home in the first leg, a margin of victory which has generally proved sufficient in the competition's recent past.
Meanwhile, Basel's 2-2 draw at Tottenham Hotspur last Thursday makes them favourites to go through at St Jakob-Park and continue their European odyssey.
Chelsea visit Moscow's Luzhniki stadium, venue of their 2008 Champions League final defeat to Manchester United, with a 3-1 lead against Rubin Kazan and a 100 percent record in their five meetings against Russian clubs.
Benfica lead by the same score against Newcastle and Fenerbahce are 2-0 ahead against Lazio, who will be playing behind closed doors at Rome's Stadio Olimpico because of racist behaviour and firework use by fans at previous home games.
Fulham and Steaua Bucharest are the only teams to have overturned two-goal, first leg deficits in the knockout stage of the competition since its name was changed from the UEFA Cup in 2009.
FC Basel, the only domestic champions among the eight quarter-finalists, will be playing their 18th European match of the season, a marathon which started against Flora in Estonia on July 17.
Since then, they have played in Norway, Romania, Belgium, Ukraine, Hungary, Russia and Portugal.
Basel's 20-year-old Egyptian Mohamed Salah was the outstanding player in the first leg, let down only by his lack of ruthlessness in front of goal.
"If Mohamed could score as well, he would not be here any more, or he would not have come here in the first place," Basel coach Murat Yakin quipped.
Tottenham, twice former winners of the old UEFA Cup, will be without Gareth Bale, Aaron Lennon and William Gallas for the match at St Jakob-Park.
Lazio, beaten UEFA Cup finalists in 1998, are unbeaten at home in the competition this season but will be playing their second home match behind closed doors as a punishment for the poor behaviour of their fans, which has marred their campaign.
Fenerbahce, whose most recent appearance in a European quarter-final ended in defeat to Chelsea in the Champions League five years ago, have lost on all six previous visits to Italy.
"I think playing without supporters is a disadvantage for any footballer, because you want to play for the supporters," Fenerbahce forward Dirk Kuyt said.
"That's the worst punishment a footballer can get. Hopefully it won't happen again but we'll be playing another silent game next week.
"The strategy for the second match will be controlling the game and not giving any chances to Lazio to score."
Rubin Kazan have already knocked defending Europa League champions Atletico Madrid out of the competition and would complete an unlikely double if they could oust the Champions League holders as well.
Chelsea are the only one of eight teams who parachuted into the Europa League from the Champions League still in the competition.
Midfielder John Obi Mikel said it was a struggle to adapt to Europe's second-string competition.
"I had never played football on Thursday nights, so it has been very hard this season," he said. "No disrespect to the Europa League but it's not the same.
"I don't even know the Europa League tune. But when you hear the Champions League anthem, you get that buzz."
Newcastle, the third of the English trio, still believe they can find a way past Benfica and overturn their two-goal deficit.
"There's a few weak spots in their defence and hopefully we can exploit that," defender Steven Taylor said.