The tie was goalless following the first leg in Portugal before Braga beat Servette 2-1 at the Stade de Geneve to qualify for the main stage of the competition.
Servette, meanwhile, dropped into the Conference League play-off round and will face Chelsea at Stamford Bridge on August 22nd in the first leg, with the second leg held in Switzerland a week later.
Servette, who last season reached the Conference League last 16, won the Swiss Cup in June, their first major trophy in 23 years.