La Liga champions Atletico suffered a shock 3-2 defeat in Greece on the opening matchday in Group A, but a repeat never looked likely at the Vicente Calderon on Wednesday.
Atletico youth team graduate Roberto Jimenez endured a moment to forget in the Olympiacos goal when he gifted Raul Garcia the opener before Mandzukic put some early misses behind him to take centre stage.
A pair of second-half headers secured the matchball for Mandzukic by the 65th minute as Olympiacos' defensive struggles from crosses dealt them continued damage and Atletico revisited the rampant form that took them all the way to last season's final
Koke looped a 17th-minute header over, but, as Atletico's intensity dropped, Olympiacos' former Barcelona defender Eric Abidal sent a rasping volley over from a half-cleared corner.
Mandzukic's touch deserted him, allowing Abidal to extinguish a promising situation for the hosts 10 minute before the break, but he made amends by slotting home when Ansaldi's curling delivery from the left found a way through Botia.
Turan darted in from the flank and away from Arthur Masuaku in the half's final minute, but Roberto was down well to thwart the Turkey international.
Botia offered a better demonstration of his defensive talents when he forced Mandzukic wide in the 59th minute and slid in to turn his shot behind.
But the forward did not have to wait long for his second, heading clinically beyond Roberto after Turan floated a delightful ball into the area.
The hat-trick came from another cross, this time of the dead-ball variety, as Roberto's touch on Mandzukic's header from Gabi's free-kick proved insufficient.
Atletico's hero departed to a standing ovation with 20 minutes to play - replacement Griezmann having an 88th-minute effort chalked off due to fellow substitute Raul Jimenez impeding Roberto from an offside position.
