An 86th-minute header from Dirk Kuyt earned Liverpool a narrow 1-0 win over Sparta Prague on Thursday, on a night when all three British clubs in action reached the last 16.
Rangers, beaten finalists in 2008, needed an equaliser at the death to prevail at Sporting Lisbon, while Manchester City's expensively assembled squad eased past Greek representatives Aris.
Liverpool enjoyed the lion's share of possession against the Czech side but, as in the first leg, they struggled to make their dominance count.
Raul Meireles blazed over wastefully from an inviting Martin Kelly cross in the first half, while French striker David Ngog was one of several Liverpool players frustrated by Sparta goalkeeper Jaromir Blazek in the second.
With extra-time beckoning, Liverpool won a corner on the left-hand side that Meireles curled into the box and Kuyt darted away from the goal-line to cushion an intelligent header into the bottom-right corner.
"It was a difficult game. We knew from the first game that they are a strong, physical side," said Liverpool coach Kenny Dalglish, after his first ever European game at Anfield as Reds manager.
"I think the boys deserve great credit for their determination and effort in getting a result. It was superb the way they took pride in themselves and the club."
Liverpool will now play Portugal's Sporting Braga, who overcame a 1-0 first-leg deficit with a 2-0 win at home to Lech Poznan of Poland.
A dramatic injury-time goal by American international midfielder Maurice Edu earned Rangers a 2-2 draw at Sporting Lisbon that sent the Scottish giants through on away goals.
After former Rangers man Pedro Mendes had cancelled out El Hadji Diouf's opener for the visitors, an 83rd-minute header by Yannick Djalo looked to have given the Portuguese team victory.
Rangers launched a desperate last attack in the 92nd minute, however, and when David Healy rolled in a low cross from the Rangers right, Edu tapped home to send the travelling fans beserk.
"We came here intent on advancing and made life a bit difficult for ourselves but we found a way to pull ourselves through in the end," Edu told the Rangers website.
The next stop for Walter Smith's side will be PSV Eindhoven, who won 3-1 at home to French league leaders Lille for a 5-3 aggregate success.
Following a goalless first leg, City beat Aris 3-0 at the City of Manchester Stadium thanks to a first-half brace from January signing Edin Dzeko and a deflected Yaya Toure effort.
Roberto Mancini's men must now tackle Dynamo Kiev, after the Ukrainian powerhouses thrashed Besiktas 4-0 to seal an embarrassingly one-sided 8-1 overall victory.
Elsewhere, Villarreal came from behind to defeat Napoli 2-1 in the tie of the round at El Madrigal.
Marek Hamsik broke the deadlock in the tie in the 18th minute before quick-fire goals from strikers Nilmar and Giuseppe Rossi just before half-time saw the Spanish side through.
Villarreal's last-16 adversaries are Bayer Leverkusen, for whom Michael Ballack scored his first goal since his return from injury in a 2-0 win at home to Metalist Kharkiv that completed a 6-0 aggregate victory.
Two-time European champions Benfica advanced with a 2-0 aggregate defeat of Stuttgart and will now face Paris Saint-Germain, who went through on away goals following a 0-0 draw at home to BATE Borisov.
In the day's early kick-off, 2008 champions Zenit Saint-Petersburg secured their place in the last 16 by coming from behind to defeat Swiss side Young Boys 3-1, thereby overturning a 2-1 deficit from the away leg.
Porto, champions in 2003, progressed on away goals on Wednesday despite a 1-0 loss at home to Sevilla, while CSKA Moscow secured a 2-1 aggregate win over Greek outfit PAOK on Tuesday.