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.
Former Spain Under-21 international Alberto Botia missed his clearance to leave the Croatia international with a 38th-minute tap in.
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
Atletico pinned Olympiacos deep inside their own half during a high-tempo opening when Juanfran and Arda Turan enjoyed plenty of joy down the right flank.
Mandzukic twice came close to finding a decisive touch on the end of low crosses before their pressure told in the ninth minute.
Juanfran and Turan combined for the latter to fire in another low centre that was blocked, but Roberto inexplicably cleared with his feet, straight back to the midfielder, who made no mistake this time in picking out Raul Garcia - recalled to the starting line-up in place of France international Antoine Griezmann - to turn home.
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.
Roberto partially atoned for his earlier error by turning a skidding low drive from Cristian Ansaldi around his left post.
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.
A muscle strain meant Atletico midfielder Tiago was replaced by Mario Suarez for the second half, while visiting coach and ex-Real Madrid star Michel sent on David Fuster and Pajtim Kasami - the latter for ineffective Barcelona loanee Ibrahim Afellay - in a bid to turn the tide.
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.
Michel will hope Atletico display similar form in their final group game against Juventus, as a victory over Malmo would send Olympiacos through in second place if the Italian side are defeated.
