The hosts took the lead on 19 minutes at the Georgios Karaiskakis Stadium through Kostas Fortounis, who rifled a free kick from 25 yards over the Clarets' wall and into the bottom corner of the net.
Burnley held out until the break but the Greek giants restored their lead on 48 minutes through another set piece, Andreas Bouchalakis heading home from Fortounis's free kick.
Things went from bad to worse for the visitors on the hour as a Ben Gibson handball in the box led to a sending-off via a second booking and a penalty for Olympiacos, which Fortounis confidently slotted into the bottom corner.
Burnley, to their credit, went up a gear in the latter stages and thought that they had grabbed a goal back on 88 minutes, only for Sam Vokes to see his close-range tap-in disallowed by the flag.
The return leg takes place at Turf Moor next Thursday night, with Burnley able to reach the group stages with a 2-0 win or force extra time with a 3-1 victory.
OLYMPIACOS (4-2-3-1): Gianniotis; Elabdellaoui, Miranda, Vukovic, Tsimikas; Camara, Bouchalakis; Christodoulopoulos (Ansarifard 84'), Fortounis (Fetfatzidis 89'), Podence; Guerrero
BURNLEY (4-4-2): Heaton; Bardsley, Long, Gibson, Ward (Lennon 66'); Gudmundsson (Vokes 76'), Hendrick, Cork, Taylor; Barnes, Wood (Tarkowski 62')