A fast start by last season’s beaten finalists was rewarded when Raul Garcia prodded home at the near post after an awful attempted clearance from ex-Atletico goalkeeper Roberto.
Mandzukic had suffered a broken nose when Olympiakos beat Atletico 3-2 on matchday one, but the Croatian had a far more enjoyable evening this time round and opened his account with a tap in after Alberto Botia failed to cut out Cristian Ansaldi’s low cross.
And he completed his hat-trick with two powerful headed finishes in three minutes just after the hour mark.
Victory for Juventus away to Malmo means Atletico just need to avoid defeat by two or more goals when they travel to Turin in two weeks’ time in order to top the group.
Olympiakos, meanwhile, now need a big favour from Atletico if they are to progress as they need to beat Malmo at home next time out and hope Diego Simeone’s men win in Italy.
The visitors hadn’t played a competitive game in 17 days as the Greek league was suspended following the recent international break due to a violent attack on a senior official of the country’s referee board.
And they looked decidedly rusty as Atletico opened them up time and again down the Spaniards’ right in the opening stages.
The first of many forays forward by right-back Juanfran caused panic in the Olympiakos area after just two minutes but neither Garcia or Mandzukic could bundle the ball home.
Mandzukic was then just inches from connecting with another low cross from Juanfran to the near post.
However, Olympiakos failed to learn their lesson as the opener arrived in similar circumstances as Roberto fired an attempted clearance straight at Juanfran’s feet and this time his cross was converted at the near post by Garcia.
Roberto redeemed himself somewhat by getting down low to his left to turn Ansaldi’s fierce drive behind and Tiago should have down better when he fired well over from a well-worked free-kick.
A lack of pace cost Mandzukic moments later when he was played in behind the Olympiakos defence before being crowded out by a bunch of recovering defenders.
However, he was presented his first goal of the evening when Botia swiped at fresh air from Ansaldi’s low cross and the Croatian slotted home at the far post.
The former Bayern Munich striker was first to threaten after half-time as well as his goalbound effort was deflected behind by Alejandro Dominguez.
Mandzukic was perfectly placed again to make it 3-0 just after the hour mark when he powered a header into the bottom corner from Arda Turan’s inviting cross.
Moments later he sealed his first hat-trick in Atletico colours with another header from Gabi’s free-kick that had too much power for Roberto.
Mandzukic was then withdrawn to a hero’s ovation from the Vicente Calderon crowd as Antoine Griezmann entered the fray.
And the Frenchman thought he had added a fifth goal when he headed home at the far post from Koke’s corner, but referee Wolfgang Stark ruled it out for a foul on Roberto, much to the dismay of the home fans.