The home side were the first team to create a chance when Aleksandar Mitrovic was played through on goal, but defender Dimitris Siovas did well to get back and make an effective tackle having been beaten the first time.
It was the away side that took the lead in the 17th minute as Javier Saviola made the most of a defensive error to play in fellow striker Mitroglou who finished neatly.
Anderlecht had the chance to equalise from the spot. However, the penalty was saved by Olympiacos keeper Roberto who got down to keep out Mitrovic's effort, which was headed for the bottom-left corner.
Young midfielder Youri Tielmans, who became the first ever player born in 1997 to play in the Champions League, looked threatening for Anderlecht and he played in Dennis Praet, whose effort flew over the crossbar as Olympiacos held on to their 1-0 lead at half time.
After the break Roberto again made two strong stops to deny Anderlecht the chance to equalise, his most impressive save coming from a Mitrovic shot in the 53rd minute.
The visitors made the most of their keeper's impressive work and doubled their advantage minutes later as Mitroglou grabbed his second of the night by slotting past Thomas Kaminski.
Mitroglou went on to seal the away side's victory as he grabbed his hat-trick with a strike which flew into the top corner.