Eager to put Sunday's (AEDT) 2-0 Premier League loss to Tottenham behind them, Pep Guardiola's men controlled proceedings in Thursday's Group C clash in Athens.
But they had only one goal to show for it as Phil Foden, who wowed for England during the international break, swept the visitors into a first-half lead that never came under threat from the toothless hosts.
Guardiola, who signed a new contract last week, saw his side frustrated by a combination of profligate finishing and fine goalkeeping from Jose Sa.
City, which scored nine goals across its opening three group games, was quickly on the front foot and Foden stung the palms of Sa with a low drive, with the goalkeeper also gathering a Gabriel Jesus header.
Jesus came far closer to breaking the deadlock on the half-hour mark, racing in behind before seeing his shot blocked away by the defiant Sa.
The opener finally arrived in the 36th minute, Foden converting from Raheem Sterling's back-heeled assist after City's slick play had Olympiacos chasing shadows.
Sterling had the busy Sa scrambling across his line to repel a dipping free-kick in first-half stoppage time, with City denied again after the interval as Ilkay Gundogan's effort flashed wide and Joao Cancelo's flick was saved.
Olympiacos was simply not able to apply any pressure – their first shot did not arrive until the 88th minute – leaving City to coast to a win that was brightened further by the late introduction of Sergio Aguero from the bench after an injury lay-off.