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.
Olympiakos struck back just six minutes later through Kostas Mitroglou which helped them to maintain second place in Group H with six points, one behind leaders Arsenal who drew 1-1 at AZ Alkmaar.
The Dutch side leapfrogged Standard Liege into third spot.
"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.
Olympiakos reacted immediately through Mitroglou, who scored his first group stage goal with a looping header over Sinan Bolat after connecting with a Luciano Galletti freekick.
Both teams continued to chase victory in an absorbing affair as Mitroglou tried his luck with an over-head kick before Jovanovic cracked a stunning long range effort off the crossbar.
Olympiakos poured forward in the later stages with Mellberg shooting straight at Bolat in the 75th minute.
Dudu had the ball in the Liege net after 84 minutes but his effort was ruled out for handball but Stoltidis pounced in the dying seconds.
Liege coach Laszlo Boloni said his team would learn some important lessons from the match.
"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."
