And the visitors, making their Champions League debut, silenced the home support with a goal from such a 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 a free kick narrowly wide and missed with a header as Celtic pressed with increased passion but a lack of guile remained in 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.
Celtic continued to attack until the final whistle but the best chance in the final stages fell to Braga and Alan lifted a shot over the crossbar from a promising position.
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