Elsewhere there were wins for Barcelona, Lyon, while Inter Milan, Stuttgart and AZ Alkmaar were held to draws.
Liverpool were in irresistible form against Hull City in the Premier League at the weekend but their defensive shortcomings were exposed in a 2-0 defeat to Fiorentina at Artemio Franchi.
Stevan Jovetic latched onto a fine pass from Cristiano Zanetti to slot home Viola's opener after 28 minutes and the Montenegro international grabbed his second of the evening with a clever flick nine minutes later.
The Reds improved after the break but Fiorentina withstood long periods of pressure to join them on three points in Group E, three points behind leaders Lyon.
Hungarian champions Debrecen showed they were capable of competing with Europe's elite when they narrowly lost to Liverpool at Anfield but they were crushed 4-0 by Lyon in their opening home game.
Kim Kallstrom fired the French giants ahead after just three minutes and Miralem Pjanic doubled their advantage with an exquisite free-kick 10 minutes later.
Sidney Govou's header deflected off Peter Czvitkovics to make it 3-0 and Bafetimbi Gomis added a fourth early in the second half to seal a convincing win.
Arsenal's progression to the last 16 was expected to be straightforward after the draw for the group phase was made but, having been given a major scare by Standard Liege in their opening game, Olympiakos provided equally stiff resistance at Emirates Stadium before eventually going down 2-0.
The Gunners put together a number of typically slick, eye-catching moves in the first half but they could not find the finishing touch as Olympiakos, who had lost on their previous eight visits to England, somehow managed to stay level.
The second half followed a similar pattern as Greek goalkeeper Antonios Nikopolidis pulled off several stunning saves before Robin van Persie finally tapped home following good work by Eduardo in the 78th minute.
Andrey Arshavin's cheeky backheel from Cesc Fabregas' cross in the 86th minute made the result safe and took Arsenal three points clear at the top of Group H.
Arsenal's position was strengthened when Moussa Traore salvaged a 1-1 draw for Standard Liege against AZ Alkmaar with a goal in the 90th minute.
Mounir El Hamdaoui's clever finish on 48 minutes had given the Dutch side an advantage and it looked like proving decisive until the last-gasp leveller.
CapitulationRangers slipped to a 4-1 defeat against Sevilla after a second-half capitulation at Ibrox.
Walter Smith's men were denied a penalty towards the end of the first half when Steven Naismith appeared to be brought down by Abdoulay Konko, but the referee waved play on.
Sevilla were ruthless in making the most of their reprieve and in a cruel twist of fate it was Konko who put them ahead with a header from Jesus Navas' cross on 50 minutes.
Adriano slotted home a second for the Spanish side 14 minutes later and Luis Fabiano and Fredi Kanoute inflicted further damage with two goals in the space of three minutes (72 and 74).
Nacho Novo netted a consolation goal for Gers with two minutes remaining, but the heavy loss leaves them with a lot of work to do in their remaining four games.
Romanian minnows Unirea Urcizeni battled back to hold Stuttgart to a 1-1 draw in the other game in Group G.
Serdar Tasci opened the scoring for Stuttgart with a close-range finish after just five minutes, but Dacian Varga drew Unirea level three minutes after the interval.
Barcelona took control of Group F with a 2-0 victory over Dynamo Kiev at Camp Nou.
Lionel Messi broke the deadlock on 26 minutes with a trademark run and shot from the left and Pedro doubled their advantage with a fine strike 14 minutes from time.
Inter Milan's winless sequence of results in the competition continued with a 1-1 draw against Rubin Kazan in the day's early kick-off, although Jose Mourinho's side will have been pleased to claim a point from a difficult trip to Russia.
The hosts took the lead courtesy of an excellent goal from Alejandro Dominguez after 11 minutes, but Inter equalised through Dejan Stankovic's header and held on after Mario Balotelli had been sent off for a second bookable offence early in the second half.