Knowing a 2-0 win on the night would be enough to send them through, Sean Dyche's Clarets created some great chances in the first half but were unable to take them, with Sam Vokes sending one header wide and another over.
Ashley Barnes then struck narrowly wide, Vokes sent a curling effort against a post early in the second half and teenager Dwight McNeil brought a good save out of Andreas Gianniotis.
Olympiakos subsequently snatched a goal in the 83rd minute through Daniel Podence before substitute Matej Vydra, making his Burnley debut, replied with a close-range finish three minutes later.
While there was certainly encouragement to be taken from the performance, there will also surely be feelings of frustration on Burnley's part as their first taste of European football since 1966-67 concluded after three two-legged qualifying ties.
Dyche's men, who have one point from their opening three Premier League games, are back in action at Turf Moor on Sunday when they face Manchester United, and Red Devils boss Jose Mourinho was watching from the stands on Thursday evening.
As in the first leg, Burnley wore black armbands as a mark of respect for Jimmy McIlroy, the club great who died last week aged 86 and whose funeral service takes place at the stadium on Friday.
The starting line-up showed seven changes from Sunday's 4-2 loss at Fulham, with 18-year-old McNeil making his first start as he came in on the left wing as a replacement for the injured Johann Berg Gudmundsson.
There was an early scare for the hosts when McNeil lost possession and Olympiakos broke forward, but Podence could only send a shot tamely wide having entered the box.
Burnley started to find some rhythm after that, and Vokes had a penalty appeal waved away by referee Viktor Kassai, hooked a shot over the bar and then twice failed to convert from great positions.
The first instance saw him head wide as he connected with McNeil's cross and he then headed too high when the ball came to him following a corner.
Some pressure from Olympiakos followed, with Podence hitting an attempt into the side-netting and another over, either side of Miguel Guerrero shooting into Tom Heaton's arms.
Burnley's woes in front of goal then continued shortly before the interval as Vokes teed up Barnes, whose shot went just wide.
In the third minute of the second half Vokes went the closest yet as his shot from outside the area came back off the post and five minutes later Gianniotis did well to tip McNeil's firm strike over.
Burnley continued to apply pressure without finding the required killer instinct before an Olympiakos counter-attack saw the ball worked to Podence, who cut inside Long and fired past Heaton.
Vydra, having come on for Barnes in the 65th minute for his Burnley bow, then bundled the ball over the line at the other end during a scramble following a corner, giving the home fans something to cheer before Kassai's final whistle confirmed the club's European adventure as over soon after. PA