Defeat for the Greek giant would have all but quashed its hopes of making it through, but Jose Luis Mendilibar's team delivered at Stadio Georgios Karaiskaki.

Costinha put them in front with a powerful header from Rodinei's corner inside two minutes.

Though Leverkusen piled forward in search of an equaliser, Mehdi Taremi – a runner-up with Inter last season – doubled Olympiacos' lead in first-half stoppage time.

Kasper Hjulmand's Leverkusen kept up the pressure after the break, though it only got one of its seven attempts on target.

Olympiacos moved up to 22nd ahead of facing fellow play-off hopefuls Ajax in a crunch clash next week, while Leverkusen – who take on Villarreal in their final league-phase fixture – are a point ahead in 19th place.