And they may have to turn the tie around next week without star winger Aiden McGeady, whose Parkhead career could be over after he was left out of the squad for the club's first competitive game of the season.
Life after McGeady does not look very promising on this evidence, with goals from Alan, debutant Uwa Elderson Echiejile and substitute Matheus condemning Neil Lennon to defeat on his European debut as manager.
Celtic failed to muster a single shot on target in sweltering conditions, with their opponents comfortably holding sway.
Lennon insisted before the game he would field winger McGeady - who is rumoured to want to leave - tonight unless he was ordered not to pick him by the club's hierarchy.
Instead, four players were handed their competitive debuts, with summer signings Joe Ledley, Charlie Mulgrew, Efrain Juarez and Cha Du-Ri all in the starting XI.
Braga, whose summer recruitment drive dwarfed that of their opponents following the loss of some key men, handed debuts to Rodrigo Lima, and Leandro Salino.
The hosts were playing their first ever Champions League game having finished second in Portugal last season.
Mulgrew made a bad early impression in a Celtic jersey with a fourth-minute yellow card for time-wasting at a throw-in.
That was after Ki had sent a free-kick over the crossbar, won by lone frontman Georgios Samaras.
The heat, despite Lennon's confidence yesterday, was stifling and both sides looked a yard off the pace in what was their maiden competitive game of the season.
Indeed, there was no real goalmouth action until the 17th minute, when a cross from Shaun Maloney - out since October due to injury - was acrobatically cleared by Moises.
Braga broke up the other end and Lukasz Zaluska spilled a Paulo Cesar shot he should have held before Andres Madrid blazed his follow-up over from 20 yards.
Then, in the 25th minute, the hosts were awarded a controversial penalty when Ki was adjudged to have handled Miguel Garcia's cross, earning a booking.
Alan fired the spot-kick into the roof of the net to put Braga ahead.
Salino could have immediately made it 2-0 but he scuffed his finish wide on the turn.
The already vocal home support was now bouncing - literally - while the small band of travelling fans were left to contemplate yet another reminder of Celtic's miserable European away record.
Captain Scott Brown was flagged offside attempting to volley Mulgrew's back-post cross but it was Braga who were getting closer to legally getting in behind their opponents.
Jos Hooiveld picked up a yellow card for clattering into Lima seconds after a talking to from referee Serge Gumienny.
The signs were not good at the start of the second half when Elderson got in behind Cha all too easily but his cross was too close to Zaluska.
The Celtic defence then dawdled as Cesar tried to force his way into the box, a gaggle of players doing just enough to stifle his finish.
Glenn Loovens became the latest Hoops defender booked after fouling Lima as the visitors' problems continued beyond the hour mark.
None of the new faces were making an impact as Celtic struggled to string more than a handful of passes together.
Lima was eventually taken out of the firing line, with Matheus coming on up front.
Moises had time to flash a header wide before Lennon responded by introducing new boy Daryl Murpy for Maloney.
Celtic soon won their first corner, which came to nought, but their search for an equaliser risked exposing them at the back.
But there was no legislating for the appalling defending that gifted Braga their second goal 15 minutes from time, Matheus' corner missed by a host of players before Elderson bundled the ball home.
Lennon threw on Forrest for the disappointing Juarez knowing an away goal would change the whole complexion of the tie.
Charlie Mulgrew sent a free-kick wide which Mario Felgueiras had covered before Matheus showed him how it should be done.
The substitute stepped up after a foul three minutes from time and blasted a shot which flew into the top corner.
Salino became the only Braga player booked in stoppage-time and Alan was withdrawn for Helder Barbosa as the home fans celebrated.