Burnley – who played the last 30 minutes with 10 men after Ben Gibson's red card – still retain a glimmer of hope thanks to Chris Wood's penalty, but they will need to turn on the style at Turf Moor in seven days' time if they are to progress to the group stage.
Fortounis opened the scoring with a superb free-kick before setting up Andreas Bouchalakis for its second after Wood had restored parity.
A penalty from Fortounis completed the victory for the 44-time Greek champions and Burnley were grateful to Tom Heaton for keeping the deficit at just two goals.
After a bright start the Premier League side found themselves on the back foot as Olympiacos settled into a rhythm.
Its pressure eventually told as Fortounis broke the deadlock, the forward's stunning effort from distance too good for the recalled Heaton.
Olympiacos threatened a second just before the half hour but they failed to take it and were duly punished, Wood slotting home from 12 yards after he had been fouled by Mohamed Camara.
Burnley was chasing the game again early in the second half, though, as Bouchalakis got in front of Jack Cork to head home Fortounis' dangerous free kick.
Its task was dealt a further blow when Gibson was shown a second yellow card for handling Camara's shot inside the area, Fortounis making no mistake from the spot.
Daniel Podence and Miguel Angel Guerrero both wasted gilt-edged opportunities to put the tie to bed but their profligacy will give Burnley the feintest of hopes.