The Serbian saw his 11th-minute spot-kick saved by goalkeeper Quim and the Portuguese hosts went on to claim victory thanks to a touch of fortune as Hugo Viana's 51st-minute strike took a huge deflection off Curtis Davies.
The result sees Braga qualify for the final 32 from a tight Group H based and means Blues must now defeat NK Maribor in the final round of fixtures and hope Club Brugge and Braga do not play out a draw, which would see the Belgian side through.
Wade Elliott and Chris Burke enjoyed success in the early stages down the right channel and it was from that area that the former won a penalty for Blues when he was brought down by Ewerton on the edge of the box. However, Zigic saw the golden opportunity slip from his grasp as Quim dived to his left to produce a parried save.
Braga steadied the ship after the early pressure and kept possession well, although only Marcio Mossoro went relatively close with a curled effort.
Helder Barbosa passed up a great chance to test Boaz Myhill as half-time approached but he completely fluffed his attempted shot from Alan's low pass.
Keith Fahey could have made the Portuguese side pay at the other end but saw his low drive well swallowed up by Quim on the stroke of half-time.
Yet just minutes after half-time Chris Hughton's side saw all their good work unravelled thanks to a chunk of misfortune. Viana, formerly of Newcastle, received the ball 30 yards out and his left-footed effort took a huge deflection off the unfortunate Davies and wrong-footed Myhill.
Barbosa was again guilty of missing a great opportunity when he fluffed his lines for a second time, on this occasion when trying to convert Mossoro's pass.
Marlon King and Burke got in each other's way when either man could have scored the equaliser while 10 minutes from time Rodrigo Lima was clean through on goal but denied by Myhill, with Paulo Cesar somehow spooning the rebound over the bar.