Midfielder Stoltidis, on as a second half substitute, rose unchallenged following a corner in the third minute of injury time to nod in the winning goal following a breathless finish.
The visitors took the lead in the 37th minute when Igor De Camargo slotted past Antonis Nikopolidis from inside the area.
"We are very happy because we played like a big club and we played with composure," said Olympiakos coach Zico.
"We went a goal down but the players... kept their heads and that allowed us to turn around the result in win the game."
Olympiakos's Jaouad Zairi and Standard midfielder Wilfried Dalmat both squandered chances in the first half.
Olympiakos enjoyed most of the possession but Standard opened the scoring through De Camargo.
LOOPING HEADER
The Liege skipper, who passed a late fitness test after recovering from a broken toe, ran on to Milan Jovanovic's clever through ball and slotted past Nikopolidis in the 37th minute.
"It was an unfair result for me," said the Romanian. "We played well but not good enough to take the victory.
"I am very disappointed that we have come away empty handed but Olympiakos were simply better going forward. We are still a new team so we can learn a lot from these games and use our experience for the future."
