Olympiacos are top of Group A after former Fulham midfielder Pajtim Kasami grabbed the winner in a 1-0 victory over Juventus.
The Greek champions are level on points with Atletico Madrid, who sit second after hammering Malmo 5-0 at the Vicente Calderon.
All the goals came in the second half as Koke, Mario Mandzukic, Antoine Griezmann, Diego Godin and Alessio Cerci secured a valuable three points for the Spanish champions.
Giulio Donati and Kyriakos Papadopoulos both scored in the second half as Bayer Leverkusen beat Zenit St Petersburg 2-0 to take charge of Group C.
Monaco climbed above Andre Villas-Boas's side into second place as they drew 0-0 with Benfica, who earned their first point of the campaign. Borussia Dortmund hammered Galatasaray 4-0 away from home to make it three wins out of three in Group D.
Pierre-Emerick Aubameyang scored twice in 18 minutes for the visitors, first tapping home Marco Reus’ pin-point pass before adding another from close range.
Reus put the Greek champions out of sight before half-time with a stunning, curling shot before substitute Adrian Ramos added a fourth late on. Arsenal look to be joining Dortmund in the knock-out phase after two goals at the death secured a remarkable 2-1 win at Anderlecht.
Anderlecht were on course for a shock victory when Andy Najar found the net with a brilliant header in the second half but there was late drama as Kieran Gibbs side-footed home an equaliser in the 89th minute before Lukas Podolski completed the turn-around with an emphatic finish in injury-time.
The win puts leaves Arsenal second, five points ahead of their opponents and three behind Dortmund.
Real Madrid delivered a first-half master-class to thrash Liverpool 3-0 at Anfield and go six points clear at the top of Group B.
The two sides have lifted 15 European Cups between them but the contest was all-but over before the break as Madrid stunned the hosts with three goals in 18 minutes.
Cristiano Ronaldo broke the deadlock in the 21st minute, latching onto James Rodriguez's scooped ball through and firing a superb half-volley past Simon Mignolet for his 70th Champions League goal. It was one-way traffic from thereon in as Karim Benzema headed Tony Kroos's cross into the far corner before the French striker poked in a third after Liverpool failed to clear a corner.
Liverpool’s chances of qualification are far from over, however, as fellow challengers Basel lost 1-0 at Ludogorets.
Basel’s Serey Die was sent off in the 18th minute but the game looked to be heading for a goalless draw until defender Yordan Minev scored the decisive goal, meaning Ludogorets, Liverpool and Basel have three points each.
