The Gunners responded from the disappointment of defeat by Premier League leaders Chelsea as goals in each half from Gervinho and Lukas Podolski and a well-taken late goal from substitute Aaron Ramsey made it successive victories in Group B.
Another three points against Schalke, who drew at home to Montpellier tonight, here in a fortnight's time would all but seal progress and probable seeding for the last 16.
With manager Arsene Wenger watching in the stands for the second of his three-game European ban, the Gunners started brightly as the rain lashed down on north London.
Gervinho was again deployed down the middle, ahead of England forward Theo Walcott and France international Olivier Giroud, who both watched on from the bench.
Both full-backs, Kieran Gibbs and Carl Jenkinson - potential call-ups for England tomorrow - made ground down the flanks as the Greek champions were stretched.
In the 11th minute, goalkeeper Balazs Megyeri tipped over Santi Cazorla's curling free-kick.
Olympiacos showed some intent, if without really stretching the Arsenal rearguard.
Gervinho appealed for a penalty when his flick caught Kostas Manolas on the arm, but Norwegian referee Svein Oddvar Moen was having none of it.
Laurent Koscielny, poor against Chelsea on Saturday, was shown a yellow card for clattering into Paulo Machado, which on another night could well have been red.
Arsenal keeper Mannone had not had much to do in the first half hour, but was alert enough when Machado's cutback was turned goalwards across the wet surface by Kostas Mitroglou and the Italian got down to grab the ball as it headed towards the bottom right corner.
Olympiacos should have been in front in the 34th minute when captain Giannis Maniatis skipped down the right and crossed through the six-yard box, but arriving at pace Machado could only scoop the ball high over the crossbar.
Against the run of play, the Gunners were ahead three minutes before half-time.
Gibbs and Lukas Podolski worked space down the left, and the ball was played back across the edge of the penalty area. Mikel Arteta won a 50/50 with Pablo Contreras, which saw possession break to Gervinho, who sidestepped his marker before beating the wrong-footed Megyeri with a low shot.
However, the lead did not last to the interval.
Arsenal were again caught cold at the back as Italian Leandro Greco whipped over a superb cross from the left touchline, which was guided into the net by a deft header from Mitroglou, who got between the two centre-halves.
There was more urgency about the Gunners after the break. Cazorla's fired wide after good work from Gervinho, before home pressure was rewarded in the 56th minute.
Again Gervinho was involved as he battled into the left side of the penalty area and picked out Podolski, who turned to drill the ball through the keeper's legs.
With confidence restored, Arsenal maintained the tempo as Koscielny headed over.
Walcott was sent on to give the Greeks something else to worry about for the final 20 minutes, replacing Alex Oxlade-Chamberlain.
More changes followed as goalscorers Gervinho and Podolski were given a well-earned break as Giroud and Ramsey came on.
Walcott darted down the right and crossed into the six-yard box, where Giroud's deft touch unfortunately floated straight to the keeper.
The Frenchman - signed for £12million from Montpellier in the summer, but with only one goal in the Capital One Cup against Coventry so far - was out of luck again in the closing stages as his first-time shot from Cazorla's cross his Dimitrios Siovas at point-blank range.
Ramsey wrapped up the victory when he latched onto Giroud's flicked header from Mannone's long goal-kick to skip into the penalty area and calmly chip the keeper.