Braga won 1-0 on the night in Turkey but the losing finalists from last season bowed out of the Europa League as they suffered a 2-1 aggregate defeat against Besiktas.
Rodrigo Lima's goal got Braga back into it midway through the second half, but Besiktas managed to keep them on the back foot for much of the rest of the match before they squandered a host of late chances.
Needing an early goal to get back into the tie, Braga wasted little time in setting about the Besiktas goal, with Alan Osorio denied early on before Hugo Viana sent a long-range attempt wide and Ruben Amorim's close-range effort went over.
They got the breakthrough in the 25th minute, with Lima latching on to Alan Osorio's pass to slot home from inside the six-yard box.
Besiktas responded with a Veli Kavlak effort from 25 yards that went wide before Ricardo Quaresma tested Silva Quim from the edge of the area.
The hosts were very much back on top, stifling Braga's attempts to force a second by pegging them back in their own half.
Hugo Almeida had a shot charged down by Ferreira Douglao and Quim could only parry Manuel Fernandes' effort wide for a corner.
Douglao got in another block to deny Almeida while, at the other end, Lima fired wide as Braga had the last chance of the first half.
Besiktas were on top again early in the second half, but Quaresma could not keep his shot down and Tomas Sivok's close-range effort from a corner was turned away by Quim.
As the game moved into the final 20 minutes, Braga pushed forward again but Lima missed the target twice in quick succession while Paulo Cesar's effort was charged down by Sivok.
Besiktas came back as Quim denied Fernandes and Almeida, while Quaresma's shot was blocked.
Braga's efforts became more frantic with time ticking down, and Douglao blazing over from close-range and Osorio denied just yards in front of goal as Ibrahim Toraman made a critical block.
Cenk Gonen held Viana's 89th-minute shot and then denied Custodio in injury time as Besiktas hung on to the win.