With the Greek side leading thanks to Pajtim Kasami's first-half goal at the Karaiskakis Stadium in this UEFA Champions League Group A encounter, Spaniard Roberto produced a string of superb saves after the interval to ensure they claimed all three points.
In the space of 15 minutes midway through the second period, he made remarkable stops to deny Alvaro Morata, Paul Pogba and Carlos Tevez on what was a frustrating night for Massimiliano Allegri's men.
Former Fulham man Kasami had earlier struck a low finish into the bottom left-hand corner as Olympiacos continued their strong recent run of results at home on Europe's biggest club stage.
Over the past 12 months, Olympiacos have beaten Benfica, Anderlecht, Manchester United and Atletico Madrid on their own soil, and this result against the Italian champions represented their fifth consecutive home victory in the Champions League.
Juve, who dropped points in Serie A for the first time this season on Saturday with a 1-1 draw at Sassuolo, had Chile international Arturo Vidal back in the side after he was rested at the weekend, but the midfielder failed to have a telling impact as Olympiacos held on to move onto six points at the halfway stage in the group, three more than Juventus.
The hosts made a bright start, Kostas Mitroglou flicking a half-chance wide before Alejandro Dominguez brought a flying save out of Gianluigi Buffon with a free-kick in the opening seven minutes.
Juve went close themselves five minutes later as Tevez's stabbed effort was headed off the line by Omar Elabdellaoui as Olympiacos failed to deal with Andrea Pirlo's corner.
Tevez did have the ball in the net in the 20th minute, but his header was rightly disallowed for offside as the hosts showed further hesitancy from a Pirlo set-piece.
The game continued to ebb and flow, and a superb touch and volley from Dominguez flashed across the face of goal in the 27th minute.
Ten minutes before the break, Olympiacos opened the scoring with a well-worked goal.
Dominguez led the break and played the ball to Mitroglou, who laid a pass back into Kasami - the midfielder firing left footed beyond Buffon from the edge of the penalty area.
Morata went close with a left-footed volley six minutes after the interval.
Kasami should have doubled his and Olympiacos' tally on the hour mark, but he rolled the ball wide from six yards after being teed up by the influential Dominguez.
Juve almost took advantage of that miss five minutes later, but Roberto brilliantly parried Morata's thumping effort from eight yards.
Roberto performed heroics once more as the game entered the final quarter of an hour, clawing away Pogba's cross before keeping out Tevez's close-range effort with his legs.
With nine minutes remaining, he pulled off another outstanding save, reacting sharply to tip Morata's fierce drive onto the crossbar as Juve continued to press hard for an equaliser.
But a leveller did not arrive as the Greek side held on for a battling victory that puts them in pole position to qualify for the knockout stages.
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