Ferran Torres wrote his name into the history books to help Manchester City avoid what could have been a Greek tragedy at the Etihad.
Torres struck on 12 minutes to become the third youngest player in the Champions League era to score in four successive games after Kylian Mbappe and Erling Haaland.
The winger has made an impressive start to life in Manchester since joining City from Valencia in a £21m deal this summer.
And he enhanced his growing reputation ever further with a neat finish from Kevin De Bruyne’s pass, before substitutes Gabriel Jesus and Joao Cancelo sealed the win with impressive late strikes as Pep Guardiola’s men made it a hat-trick of wins in Group C.
But this was far from vintage from the hosts and the margin of victory should have been so much greater for City, who dominated for long spells but saw the Greek champions miss three decent chances to take something from the game in the second half.
Guardiola made five changes from the side that beat Sheffield United last weekend and reshuffled his defence, with right-back Kyle Walker the only one to keep his place at the back.
The Spaniard chose to rest Aymeric Laporte and Ruben Dias, recalling John Stones and Nathan Ake to partner each other at centre back for the first time since the first game of the season.
Whatever combinations Guardiola put out, City were still red hot favourites to win and take a giant step towards the knockout stages.
Olympiakos were no strangers to these shores, but on 17 visits to England the Greek outfit had tasted victory just twice, both of them coming against Arsenal.
To make matters, only Barcelona and Porto had conceded more away Champions League goals here than Pedro Martin’s men so the omens all pointed to a comfortable win for the home side.
The tone was set from the start as City took the ball and refused to give it back.
Pape Abou Cisse was forced to hack clear Phil Foden’s cross from underneath his own bar inside five minutes, before then having to block a close range shot from Riyad Mahrez.
But a breakthrough was inevitable and it duly arrived moments later when a neat interchange between De Bruyne and Torres saw the Spaniard beat Jose Sa with a calm finish to make it 1-0.
A flood of goals was expected, but somehow the visitors managed to escape more damage as City started to be wasteful and complacent going forward.
De Bruyne stung the palms of Sa and Foden shot tamely at the Olympiakos stopper, but the visitors started to grow in confidence.
Yet Mathieu Valbuena failed to capitalise on a misplaced pass from Foden, before Youssef El-Arabi twice went close to finding an unexpected equaliser.
His wastefulness was punished when two sublime strikes in the last nine minutes from Jesus and then Cancelo gave the scoreline a somewhat flattering look.
MANCHESTER CITY (4-3-3): Ederson 7; Walker 8 (Cancelo 81), Stones 7, Ake 7, Zinchenko 6; De Bruyne 7 (Nmecha 85), Gundogan 7, Foden 7 (Jesus 68, 6); Mahrez 6 (Rodri 68, 7), Sterling 7 (Silva 80), Torres 9.
OLYMPIAKOS (4-3-3): Sa 6; Rafinha 6 , Semedo 6, Cisse 7, Holebas 6; M’Vila 6, Bouchalakis 7 (Pepe 46, 7), Camara 6 (Masouras 74); Randjelovic 5 (Bruma 46, 6), El-Arabi 6 (Hassan 76), Valbuena 8 (Soudani 85).
REFEREE: Carlos Grande 7.
MOTM: Ferran Torres.