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There was plenty of attacking intent from both sides early on, with speculative forays forward and a number of prickly challenges punctuating the first few minutes of the game.
Olympiacos were responsible for the first concrete chance of the game just before the quarter-hour mark when Alberto Botia was first to a free kick, jumping high and sending a towering header inches wide of the right-hand post.
The hosts were next to threaten when, 19 minutes in, Junior Fernandes produced a ferocious strike from the edge of the box, but Roberto Jimenez pulled off an acrobatic reflex save to keep the ball out.
Both sides continued to labour in the final third as the game went on, with many efforts either flying high into the stands or safely into the arms of the opposition goalkeepers.
Arthur Masuaku tested Eduardo in the 32nd minute with a powerful strike following an incisive through-ball, while Brown Ideye wasted a great chance four minutes later when his header from near the penalty spot soared high above the crossbar.
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The action went up a gear after the restart when, 52 minutes in, Domagoj Antolic picked the ball up in space inside the penalty area and fired a shot just wide of the left post.
Moments later the Greek giants countered with an equally dangerous chance courtesy of Ideye, who collected a precise pass inside the box and sliced a shot inches off target, and not long passed thereafter before Konstantinos Fortounis weighed in with an effort which was saved by Eduardo.
A lull followed, but in the 79th minute Olympiacos succeeded in taking the lead, with Ideye finding space inside the box to meet a pinpoint cross before thumping a precise low drive into the net.
Angelo Henriquez came close to equalising a minute from time when he got onto the end of a lofted pass into the goal area, but his thumping header soared narrowly wide of the left post as the referee's final whistle confirmed all three points for the visitors.
The result sees the Piraeus-based outfit climb to second in Group F on six points and establish a three-point gap between themselves and third-placed Zagreb.