Braga's hopes of repeating last year's run to the Europa League final were hanging by a thread after goals from Tomas Sivok and Simao gave Besiktas a 2-0 victory in the first leg in Portugal.
Helder Barbosa's first half sending-off for two bookable offences hindered the home side, who must now pull off an extraordinary recovery in the second leg in Turkey in order to book their place in the last 16.
Braga went into the game on a high on the back of nine straight wins which had lifted them to third in the Liga, with Hugo Viana bending a superb free-kick just over the bar and Barbosa firing wide.
Besiktas by contrast were without a win in their last four Super Lig outings but in Portuguese coach Carlos Carvalhal, who supported and briefly managed Braga and still lives in the city, they had expert insight.
The dangerous Ricardo Quaresma came close to giving the visitors a 15th-minute lead when he curled a long-range shot narrowly over the bar, and two minutes later a weak shot in the box let Braga off.
The game changed in a five minute spell beginning in the 23rd minute when Barbosa was booked for fouling Kavlak, and dismissed in the 28th minute for a repeat offence on Ibrahim Toraman.
Besiktas grabbed the lead 10 minutes before half-time after a sustained period of pressure, Manuel Fernandes flinging in his second consecutive corner and Sivok rising to head past Braga keeper Joaquim Silva.
Braga struggled to respond, with Custudio hitting a long-range effort over the bar, Ruben Amorim also firing wide from long range and Paulo Cesar bringing a comfortable save out of Cenk Gonen from 25 yards.
Egeman Korkmaz headed just over early in the second half for Besiktas but the visitors extended their lead in the 58th minute when Fernandes fed Simao and the forward struck home from 12 yards.
Silva saved Braga from going further behind five minutes from time when he tipped a Kavlak shot round the post, while Hugo Almeida's injury-time effort was blocked in the box by Ewerton.
