Seydou Doumbia opened the scoring after just 70 seconds, before Gelson Martins took full advantage of some kamikaze defending to double the visitors' lead inside 14 minutes.
After hitting the woodwork twice, Sporting netted a third through Bruno Fernandes' clipped finish over a horribly exposed Stefanos Kapino in the Olympiacos goal.
Olympiacos produced a dramatic late rally, with substitute Felipe Pardo scoring twice, but the damage had been done in a dreadful first-half display.
Besnik Hasi's men must now pick themselves up for what will be a daunting trip to Juventus in two weeks.
Sporting, however, will have a spring in their step ahead of Barcelona's visit to Estadio Jose Alvalade.
The visitors got off to a sensational start with on-loan Roma striker Doumbia opening his account for Sporting.
Ivory Coast international Doumbia was given the freedom of the Olympiacos area to meet Marcos Acuna's teasing free-kick and head past the flailing Kapino.
A shell-shocked defence was breached again a little over ten minutes later.
An Olympiacos attack was swiftly countered by Sporting, with Doumbia releasing Martins just over the halfway line. Martins' pace took him away from the backtracking defenders to coolly slot past Kapino.
Sporting then nearly netted a third in the 21st minute. Fernandes spun elegantly on the edge of the area but his curled effort crashed back off the post.
Martins should have had his second soon after but he somehow thumped his close-range effort against the crossbar when it looked easier to score.
The Portuguese did not have to wait long to make it three, though, as they once again broke through the home defence with alarming ease.
Sebastian Coates' through ball from defence was latched onto by Fernandes, and without a defender in sight, he was able to clip over Kapino to give his side a thoroughly deserved 3-0 lead at the break.
Safe in the knowledge that they were, barring a remarkable turnaround, taking three points back to Portugal, Sporting were nothing like as potent in the second half.
In fact, it was Olympiacos who carved out the first significant chance. Uros Djurdjevic scuffed an effort towards goal, but the impressive Coates was able to divert it behind.
It was a hugely improved second half performance from the hosts, and they got their reward in the 89th minute when Pardo drilled a low shot into the bottom-left corner.
Pardo then scored his second from Diogo Figueiras' cross to set up a tense finale, but it was too little too late for Olympiacos.