Dirk Kuyt headed an 85th-minute winner to snatch a 1-0 victory over Sparta Prague in Kenny Dalglish's first European home tie as Liverpool manager.
• Second knockout round draw
• Europa League gallery Both sides had squandered numerous chances in a scrappy Europa League encounter but Liverpool's superior quality was eventually rewarded by the hard-working Kuyt.
David Ngog, Joe Cole and Raul Meireles had also gone close in front of a sell-out 42,949 crowd which further underlined the gathering momentum of Dalglish's new regime.
Sparta, backed by 3,000 of their own raucous fans, contributed fully to an eventful, if low-quality, clash but were eventually undone as Kuyt profited from a late corner. It was the only goal of the tie.
Maurice Edu scored a dramatic last-gasp goal to send Rangers through, where they will meet PSV Eindhoven, as Sporting Lisbon crashed out of the competition following a 2-2 draw at the Jose Alvalade Stadium.
El-Hadji Diouf claimed his first goal for Rangers before strikes from former Rangers midfielder Pedro Mendes and Yannick Djalo appeared to have secured victory for the home side.
But Edu pounced in injury time with the decisive goal in the tie as Rangers triumphed on away goals after the first leg had ended 1-1.
Edin Dzeko scored twice as Manchester City set up a clash with Dynamo Kiev, with an easy 3-0 win over Greek minnows Aris.
Greek minnows Aris were never in the hunt once Dzeko had made his double contribution inside the first 12 minutes. Yaya Toure's deflected effort brought City a third before the end to confirm a meeting with Andriy Shevchenko's side, the first leg in Ukraine in a fortnight.
Villarreal hit back from an early deficit to defeat Napoli and book their place in the next round following an exciting second leg at El Madrigal.
Villarreal, Spain's final representatives in the competition following Sevilla's exit last night, looked in some trouble after Napoli capped an impressive start by taking a 17th-minute lead through Marek Hamsik.
That away goal meant Villarreal, having drawn the first leg 0-0 in Italy, would need to score at least twice to avoid tumbling out, but they managed to do just that in a fourth-minute spell before the break.
Nilmar drew the teams level in the 42nd minute before Italy international Giuseppe Rossi saw his deflected shot loop in during first-half stoppage-time.
Goals by Simon Rolfes and Michael Ballack earned Bayer Leverkusen a 2-0 win on the night and a 6-0 aggregate victory over Metalist Kharkiv as they eased through to the last-16.
With Leverkusen leading 4-0 after the first leg, the game at the BayArena was in reality a formality and Jupp Heynckes' side will progress to the next round with confidence.
PSV Eindhoven progressed after fighting back from a half-time deficit to secure an ultimately convincing 3-1 victory over 10-man Lille.
PSV, who battled back from 2-0 down to draw the first leg thanks to two late goals, suffered another scare when Lille took a 22nd-minute lead against the run of play through Pierre-Alain Frau.
However, the Dutch giants fought back after the break and drew level through Balazs Dzsudzsak's quickly-taken free-kick before Lille were reduced to 10 men just after the hour when Frau picked up the second of two quick bookings.
PSV wasted no time in making the most of their man advantage as they soon added two more goals through Jeremain Lens and Antonio Marcelo to set up a clash with Rangers in the next phase.
Zenit St Petersburg came from two goals down in the tie to go through via a 3-1 (4-3 aggregate) victory over Young Boys in an exciting second leg.
The visitors led 2-1 from the first game and extended their lead via Ammar Jemal's close-range header, only for Danko Lazovic to equalise before half-time.
And that proved the turning point as Sergei Semak and Roman Shirokov struck in the second period to turn the tie around.
Basel led for 75 minutes of their Europa League tie against Spartak Moscow, but failed to add the crucial second goal before Aiden McGeady's 90th-minute equaliser won the tie outright for the hosts.
Having been beaten 3-2 at home in the first tie, Basle needed to win by two clear goals and things started well when Scott Chipperfield netted after 15 minutes.
The Swiss side could not fashion a second though, meaning their Russian opponents would have progressed on away goals before McGeady secured a 1-1 draw on the night and a 4-3 aggregate victory in the closing moments.
Ajax cruised into the next round with a 5-0 aggregate win over Anderlecht in Amsterdam.
Anderlecht general manager Herman van Holsbeeck had admitted the chances of his side overturning their 3-0 first-leg deficit were ''almost nil''.
And he was proved absolutely correct as Miralem Sulejmani scored twice in five minutes in the first half to kill the tie.
Braga overcame a first-leg deficit to see off the challenge of 10-man Lech Poznan and progress on Thursday.
First-half goals from Alan and Rodrigo Lima saw the Portuguese through to a meeting with Liverpool.
Dynamo Kiev demolished Besiktas for the second time in a week to ease into a clash with Manchester City.
Besiktas tried to put up a fight, but were ultimately undone by goals from Ognjen Vukojevic, Andriy Yarmolenko, Oleg Gusev and Andriy Shevchenko as the home side claimed an 8-1 aggregate victory to set up a clash with Manchester City.
Eredivisie side FC Twente weathered an early storm from Rubin Kazan to seal a 2-2 draw at the Arke Stadion.
Having claimed a 2-0 win in the first leg in Russia, Michel Preud'homme's side were heavy favourites to progress until Cristian Ansaldi and Cristian Noboa scored twice in the space of three minutes to set up a potential reversal.
But Twente matched their opponents' efforts in front of goal, Theo Janssen and Douglas scoring either side of half-time to tie the scores for a 4-2 aggregate victory.
Paris St Germain advanced on away goals after failing to get the better of BATE Barisov.
Peguy Luyindula's 89th-minute header earned PSG a 2-2 draw in Belarus last week and that ultimately proved crucial as the two sides fought out a goalless draw in the French capital.
Goals from Eduardo Salvio and Oscar Cardozo guided Benfica past Stuttgart 2-0 on the night and 4-1 on aggregate.
Stuttgart were looking to overturn a 2-1 deficit but it was the Portuguese side who went ahead through Salvio just after half an hour.
Benfica then made certain of their progress when Cardozo struck in the 78th minute.