Mitroglou gave Olympiakos a first-half lead which was preserved by Aleksandar Mitrovic's penalty miss.
And the Greece striker scored twice more after half-time to complete a comfortable 3-0 win which leaves his side second in Group C.
Anderlecht were forced into a late change to their line-up, with keeper Silvio Proto injured in the warm-up and replaced by Thomas Kaminski.
Both teams had early chances, with Anderlecht's Mitrovic heading Guillaume Gillet's cross back across goal and wide before Olympiakos' Andreas Samaris shot over from Vladimir Weiss' cross.
Anderlecht enjoyed a spell of pressure but Dennis Praet shot wide after a neat one-two with Mitrovic, Massimo Bruno volleyed into the side netting and Bram Nuytinck's effort was blocked by Gaetan Bong.
But it was the visitors who took the lead in the 17th minute when Javier Saviola squared the ball to Mitroglou, who finished from close range.
And Anderlecht wasted a glorious chance to level five minutes later, when Samaris fouled Sacha Kljestan in the area but Mitrovic's penalty was saved by goalkeeper Roberto.
A good cross from Yoeri Tielemans gave Praet the chance to equalise before the interval but he volleyed over.
Two Praet crosses gave the hosts hopes of an equaliser early in the second half, but Matias Suarez volleyed wide and Mitrovic was denied by a good save from Roberto.
But Olympiakos doubled their lead in the 56th minute when Nuytinck's slip allowed Mitroglou to latch on to Delvin Ndinga's header and slot the ball home.
Kljestan fired a free-kick straight into the wall and Anderlecht's hopes disappeared in the 72nd minute when Mitroglou, teed up by David Fuster, curled an unstoppable 20-yard shot into the top corner.
The hosts had no response, with Massimo Bruno, Suarez and Tielemans all shooting over, and remain pointless after two games.