The Greece striker's precision and a fine performance from keeper Roberto gave the Greek side their first points in Group C.
Anderlecht, with no points and lying fourth in the domestic league, were left to rue their misfiring front line after the Greek champions made the most of limited chances in both halves.
After several Anderlecht forays, Argentine Javier Saviola exploited disarray in the home defence in the 17th minute to find Mitroglou, who poked the ball past Anderlecht's keeper.
However, Roberto dived to his right to save Aleksandar Mitrovic's spot kick in the 24th minute.
Mitroglou nipped past a stumbling Bram Nuytinck to take a looping header in the 56th minute and completed his hat-trick in style with a long right-footed shot in the 72nd minute to knock the energy out of the Belgian champions.
"It was Mitroglou's night but I have to sing the praises of all the players," Olympiacos coach Michel said.