Both sides struggled to create goalscoring chances before the break, with neither goalkeeper being called into any telling action.
However, the visiting Greeks posed more of an attacking threat after the break and scored the only goal of the game with 20 minutes remaining courtesy of Spanish midfielder David Fuster.
Just one minute later Olympiakos keeper Balazs Megyeri was handed his marching orders for a foul inside the area on Gokdeniz Karadeniz. Carroll, making his debut, came on to palm away Bibras Natkho's spot kick.
The ten men of Olympiakos held out to take a 1-0 advantage into the return leg next week.