A second leg victory was not enough as Celtic crashed out of Champions League qualifying at the first hurdle.
Leading 3-0 from the first leg, Braga increased their advantage when Paulo Cesar glanced in a neat header midway through the first half.
Gary Hooper pulled one back from close range after the break and Efrain Juarez added a header on 78 minutes.
A stirring comeback never looked likely and Celtic now drop into the play-off round for the Europa League.
The defeat to the Portuguese means a second successive season without Champions League football.
The home crowd did their best to lift their team with a rousing reception but Celtic were insipid from the off and did not trouble the visitors at all in the first half.
Celtic were jittery in defence, gifting their opponents possession with several stray passes, and were guilty of thumping aimless long balls forward far too often.
Braga comfortable coped comfortably at the back and always looked lively on the break.
And the Portuguese silenced the home support with a goal from a lightening quick counter-attack on 20 minutes.
Alan was given room to deliver a sweet cross from the right and Paulo Cesar ran across Cha Du-Ri before steering a header beyond the outstretched arm of Lukasz Zaluska and inside the post.
Celtic striker Georgios Samaras, who was off target with an early header from a corner, later stretched his leg towards a fizzing delivery from Jaurez but could not turn the ball on target.
The hosts almost fell further behind when they were exposed down their left flank, with Matheus breaking away from Jos Hooiveld but Zaluska narrowed the angle and blocked at his near post.
Hooiveld was then beaten for pace again and was booked for pulling back Paulo Cesar on the edge of the penalty area - the resulting free-kick failing to trouble Zaluska.
Needing five goals from the second half to progress, Celtic threw on striker Marc-Antoine Fortune for left-back Charlie Mulgrew.
However, it was Braga who were first to threaten after the interval, with Cesar stealing in behind the home defence but somehow failing to pick out one of two team-mates bursting into the penalty area.
Celtic made the most of that reprieve, with Hooper, making his first appearance at Celtic Park since his move from Scunthorpe, bagging an equaliser on 51 minutes.
Fortune's power and skill won him space on the right side of the box and his excellent low cross was bundled home by Hooper.
Braga retreated deeper after conceding but Celtic needed a saving tackle by Cha to prevent Leandro Salina converting at the back-post.
Samaras curled free kick narrowly wide and missed with a header as Celtic pressed with increased passion but a lack of guile remained among the home ranks.
Neil Lennon's side went in front on 78 minutes when Juarez nodded in a cross from Samaras from a suspiciously offside-looking position.
Leading 3-0 from the first leg, Braga increased their advantage when Paulo Cesar glanced in a neat header midway through the first half.
Gary Hooper pulled one back from close range after the break and Efrain Juarez added a header on 78 minutes.
A stirring comeback never looked likely and Celtic now drop into the play-off round for the Europa League.
The defeat to the Portuguese means a second successive season without Champions League football.
The home crowd did their best to lift their team with a rousing reception but Celtic were insipid from the off and did not trouble the visitors at all in the first half.
Celtic were jittery in defence, gifting their opponents possession with several stray passes, and were guilty of thumping aimless long balls forward far too often.
Braga comfortable coped comfortably at the back and always looked lively on the break.
And the Portuguese silenced the home support with a goal from a lightening quick counter-attack on 20 minutes.
Alan was given room to deliver a sweet cross from the right and Paulo Cesar ran across Cha Du-Ri before steering a header beyond the outstretched arm of Lukasz Zaluska and inside the post.
Celtic striker Georgios Samaras, who was off target with an early header from a corner, later stretched his leg towards a fizzing delivery from Jaurez but could not turn the ball on target.
The hosts almost fell further behind when they were exposed down their left flank, with Matheus breaking away from Jos Hooiveld but Zaluska narrowed the angle and blocked at his near post.
Hooiveld was then beaten for pace again and was booked for pulling back Paulo Cesar on the edge of the penalty area - the resulting free-kick failing to trouble Zaluska.
Needing five goals from the second half to progress, Celtic threw on striker Marc-Antoine Fortune for left-back Charlie Mulgrew.
However, it was Braga who were first to threaten after the interval, with Cesar stealing in behind the home defence but somehow failing to pick out one of two team-mates bursting into the penalty area.
Celtic made the most of that reprieve, with Hooper, making his first appearance at Celtic Park since his move from Scunthorpe, bagging an equaliser on 51 minutes.
Fortune's power and skill won him space on the right side of the box and his excellent low cross was bundled home by Hooper.
Braga retreated deeper after conceding but Celtic needed a saving tackle by Cha to prevent Leandro Salina converting at the back-post.
Samaras curled free kick narrowly wide and missed with a header as Celtic pressed with increased passion but a lack of guile remained among the home ranks.
Neil Lennon's side went in front on 78 minutes when Juarez nodded in a cross from Samaras from a suspiciously offside-looking position.
