Chiquinho gave the host a lead in the eighth minute but Mbappe scored in the 22nd, 24th and 29th minutes.
Substitute Mehdi Taremi, who replaced the injured Chiquinho in the first half, hauled Olympiacos back into the match with an emphatic header past stand-in Madrid goalkeeper Andriy Lunin in the 52nd minute.
Mbappe scored Real Madrid’s fourth goal in the 60th minute and Ayoub El Kaabi halved the deficit with nine minutes remaining.