Liverpool too good for Saint-Gilloise, Brighton battle back to hold Marseille
West Ham's 2-1 at Freiburg saw them set a new record unbeaten streak for an English team in Europe, while Romelu Lukaku was among the scorers in Roma's 4-0 win over Servette.
Jurgen Klopp named a strong side at Anfield with the Reds keen to make sure of top spot in Group E and progress directly to the last 16.
Saint-Gilloise goalkeeper Anthony Moris had been in inspired form to deny Mohamed Salah early on, but his fumble from Trent Alexander-Arnold's effort left Ryan Gravernberch with a tap-in for his first Liverpool goal on the stroke of half-time.
Klopp sent on Alexis Mac Allister at half-time as he came up against his brother Kevin, who started for the Belgians.
Liverpool were made to wait until stoppage time to secure the points as Diogo Jota burst clear on goal and produced a cool finish.
The Reds travel to Brighton in the Premier League on Sunday and the Seagulls may well be fatigued after their efforts to snatch a point in southern France.
Marseille have struggled so far this season but went 2-0 up thanks to two goals in two minutes midway through the first half.
Chancel Mbemba swept the French side into the lead before a slip from Brighton captain Lewis Dunk allowed Jordan Veretout to make it 2-0.
Brighton seemed destined for a third consecutive defeat in all competitions, but the momentum swung when Pascal Gross was picked out by Kaoru Mitoma to halve the arrears.
Joao Pedro then levelled from the penalty spot after Tariq Lamptey had been upended inside the box two minutes from time to leave Brighton just one point adrift of Marseille and Ajax in Group B.
"I think we are not playing well, it's a very tough period for us in this moment," said Brighton boss Roberto De Zerbi.
"To close the first half at 2-0, this reaction is incredible. For it I am very happy."
AEK are the surprise leaders of the section on four points after coming from behind to draw 1-1 with Ajax.
West Ham set record
West Ham are well on course for the last 16 after stretching their unbeaten run in Europe to 17 games.
"It's a brilliant achievement, I've said to the boys we need to keep it going," said Hammers boss David Moyes.
"Winning the game tonight was most important."
Lucas Paqueta put the Europa Conference League winners in front after just eight minutes with a towering header.
Roland Sallai levelled for Freiburg early in the second-half.
But the Hammers were not to be denied as Nayef Aguerd headed home James Ward-Prowse's corner.
European competitions have also been Roma's strong point under Jose Mourinho and the Italians put their poor Serie A form to one side against Servette.
Lukaku opened the scoring before an Andrea Belotti brace either side of Lorenzo Pellegrini's strike made it two wins from two for Roma in Group G.
Rangers did not get any reaction from the sacking of Michael Beale as the Glasgow giants slumped to an embarrassing 2-1 defeat to Aris Limassol.
Former Gers midfielder Steven Davis took temporary charge but could only oversee Aris registering their first ever European group stage victory.
Alex Moucketou-Moussounda and Shavy Babicka scored on a famous night for the Cypriots as Abdallah Sima's header proved nothing more than a consolation for Rangers.
All four sides in Group C are now on three points after Real Betis bounced back from their 1-0 defeat by Rangers with a 2-1 win over Sparta Prague.
In the Europa Conference League, Aston Villa left it very late to see off Bosnian champions Zrinjski Mostar 1-0 as John McGinn headed in a 94th-minute winner.