The Clarets missed out on a place in the group stages of the Europa League as their heroic play-off effort came up just short.
Trailing 3-1 from the first leg in Greece, Burnley outplayed Olympiakos at a pumped-up Turf Moor as they looked to pull off a famous comeback.
They had the chances to do it in a superb performance thoroughly appreciated by some magnificent home support.
But having done everything but score, the multiple Greek champions grabbed an away goal seven minutes from the end through Daniel Podence.
Even then Burnley responded with a debut goal from substitute Matej Vydra to avoid an undeserved defeat.
But that was not enough to spare a frustrating exit and the end of a six-week run that will live long in the memory.
The Clarets, hoping to become only the third English side to successfully come back from a first-leg loss in Europa League qualifying, gave a full debut to Dwight McNeil and the 18-year-old immediately looked at home on the big stage.
He was instrumental in many of Burnley’s best moments of a first half that brought everything but a goal for the Dyche’s side as they tried to claw back the two-goal deficit.
It was a cross from McNeil that created Burnley’s clearest chance of the opening 45 minutes, Sam Vokes timing his near-post run perfectly but then steering a powerful header just the wrong side of the post.
Vokes then headed another close-range effort just over the top after the ball fizzed in his direction via the head of an Olympiakos defender following an Ashley Westwood corner.
And Westwood was also instrumental in driving Burnley forward, releasing a string of telling passes from midfielder.
Two minutes before the break such a delivery released Vokes for another run in behind the Olympiakos back line as the Wales international and strike partner Ashley Barnes caused the visitors all sorts of problems.
This time, Vokes picked out Barnes with a square pass which Barnes met, under pressure, with a first-time effort he guided agonisingly wide.
Burnley had dominated for long spells but Olympiakos always knew an away goal would give them further control of the tie and on a couple of occasions they went close to grabbing one.
Diminutive forward Podence was their chief threat on the break and after a couple of shooting chances the Clarets managed to snuff out, the Portugal under-21 international was guilty of a bad miss four minutes before half-time when he was picked out by full-back Kostas Tsimikas but lifted his shot over the bar.
But if Olympiakos lived dangerously before the break, they were hanging on in increasing desperation in the early part of the second half as the Clarets cranked up the pressure even further as the home fans urged them on in a wonderful atmosphere.
Vokes was again unlucky with a magnificent curling effort which hit a post before McNeil let fly with a stinging curling drive which goalkeeper Andreas Gianniotis was relieved to touch over the top at full stretch.
Phil Bardsley, another leading performer, then picked out Jeff Hendrick with a clever free-kick which created another chance a combination of Hendrick and Barnes couldn’t take.
And then from another opening Aaron Lennon tried to pick out Barnes when he had the opportunity to shoot and potentially give Burnley the reward they deserved to sow real seeds of doubt in the minds of the Greeks.
Dyche sent on Vydra for a debut to try and add to the goal threat and then with 15 minutes to go made the final throw of the dice by throwing on Chris Wood and Jack Cork – whose extra-time goals had taken Burnley past Aberdeen and Istanbul Basaksehir to reach this stage.
With time running out the Clarets were desperate for a goal to try and set up a grandstand finish and with eight minutes to go it almost came.
And it was Vydra with the opportunity to make an instant name for himself as goalkeeper Gianniotis palmed out a header towards the summer signing, whose first-time effort went over the bar.
And a couple of minutes later the Clarets’ magnificent effort was all but ended as Olympiacos struck on the break to book their place in the next round, Podence this time providing the finish with a strike into the roof of the net.
Still the Clarets wouldn’t lie down and Vydra claimed his first Burnley goal when he forced the ball home after a Bee Mee header had been blocked following a corner.
An equaliser was the least Dyche’s men deserved but needing two more goals to force extra time, the clock was against them as the first European campaign in 51 years came to an end on a night the Clarets gave everything they could to keep the dream alive.
Burnley: Heaton, Bardsley, Long, Mee, Taylor, Lennon (Cork 75), Westwood, Hendrick, McNeil (Wood 75), Barnes (Vydra 65), Vokes. Substitutes: Hart, Lowton, Tarkowski, Ward.
Booked: Barnes, Westwood, Bardsley, Mee
Olympiakos: Gianniotis, Elabdellaoui, Roderick, Vukovic, Tsimikas, Bouchalakis, Camara, Christodoulopoulos, Fortounis (Torosidis 87), Podence (Cisse 90), Guerrero (Hassan 71). Substitutes: Choutesiotis, Natcho, Fetfatzidis, Ansarifard.
Booked: Tsimikas, Gianniotis, Bouchalakis
Referee: Viktor Kassai (Hungary)