Goals from Gervinho, Lukas Podolski and a stoppage time effort from substitute Aaron Ramsey put a glossy sheen on a performance that for 45 minutes would have had banished coach Arsene Wenger gnashing his teeth in the stands.
The two clubs have become regular rivals in the competition in recent years, with Arsenal comfortable at home and Olympiakos enjoying superiority on home soil.
Podolski's 56th-minute effort, his second Champions League goal for Arsenal after he scored in the 2-1 victory in Montpellier two weeks ago, changed all that, knocking the stuffing out of the Greeks who reverted to their role of obliging guests whose record away to English opposition now reads played 11, lost 11.
"They made it really tough for us and defended really well," Steve Bould, standing in for Wenger who is serving a three-game UEFA touchline ban," said of a performance that epitomised the football cliche 'a win is a win'.
"We started off not bad, but then looked nervous. The result from the weekend drained us a little, we lost a little bit of confidence, but overall it is a fantastic result."
"The other game (in the group) was a draw but it's early days, we've still got an awful lot of work to do," assistant manager Bould added.
The victory margin disguised a patchy performance from Arsenal and Wenger's call for a quick response from the loss to Chelsea the previous weekend, his team's first of the season, appeared to have gone unheeded by the players as they were outshone by a creative Olympiakos side in the first half.
Former Malaga playmaker Santi Cazorla was Arsenal's main threat, the Spaniard forcing a fingertip save from Balazs Megyeri with a wickedly curling free kick, but there was precious little else to excite a home crowd enjoying a 15th successive season in the competition.
