Faced with the daunting task of overturning a 3-1 deficit against their opponents, the Clarets began brightly and won a corner in the second minute thanks to a good surge forward by Phil Bardsley.
The visitors countered with a corner of their own which was initially headed clear but subsequently recycled by Kostas Tsimikas, who blasted over the bar from distance.
The Clarets grew into the game as the minutes ticked by, Sam Vokes seeing a penalty appeal turned down and then spurning three gilt-edged opportunities before the half-hour mark.
Although Sean Dyche's side did well to contain Olympiacos, the visitors looked dangerous and, on 42 minutes, Tsimikas set up Podence in the penalty box, only for the winger to rifle over.
Neither side could break the deadlock come half time and the second half saw more frustration for Burnley, who wasted numerous opportunities to not only take the lead but potentially kill off the tie.
Indeed, the Lancastrians were punished on 83 minutes as a counter-attack culminated in Podence picking up a pass and turning inside, getting the better of Bardsley and firing into the roof of the net.
Although Burnley did eventually breach their opponents' clean sheet, with Matej Vydra stabbing in from close range after a goalmouth scramble from a corner, it proved too little too late for the Premier League club, whose 4-2 aggregate defeat means an end to their European journey.
BURNLEY (4-4-2): Heaton; Bardsley, Long, Mee, Taylor; Lennon (Wood 75'), Westwood, Hendrick, McNeil (Cork 75'); Vokes, Barnes (Vydra 65')
OLYMPIACOS (4-2-3-1): Gianniotis; Elabdellaoui, Miranda, Vukovic, Tsimikas; Camara, Bouchalakis; Christodoulopoulos, Fortounis (Torosidis 88'), Podence (Cisse 90+4'); Guerrero (Hassan 72')