First Hoops goals for Gary Hooper and Efrain Juarez proved to be the only highlights of the club's Champions League campaign after Paulo Cesar's first-half away goal killed off any slim hopes of a comeback.
Slack defending from new full-back Cha Du-ri allowed Cesar to put Braga four goals ahead on aggregate midway through a disappointing first half for Neil Lennon's men.
Hooper and Juarez salvaged some pride to win the game but Celtic never built up the attacking momentum needed to prevent them dropping into the Europa League play-off round.
Celtic manager Neil Lennon handed striker Hooper a competitive debut, the introduction of the recent signing from Scunthorpe the only change to the team that lost 3-0 in Portugal in the first leg of the third qualifying round.
Midfielder Ki Sung-yueng made way for Hooper, whose goalscoring appearance against Lyon on Saturday was his first pre-season match action.
With Aiden McGeady considering a move to Spartak Moscow, Celtic started with Shaun Maloney and Efrain Juarez out wide.
They twice threatened from inswinging corners by Charlie Mulgrew in the opening stages.
However Georgios Samaras glanced his header wide from the first and the second broke to Joe Ledley, who sliced over from 16 yards.
But the early pressure soon dissipated and Celtic's task, already tough, grew even more difficult when Braga snatched an away goal in the 21st minute.
Alan was allowed room to cross from the right and Cesar ran across Cha before glancing a header beyond the outstretched arm of Lukasz Zaluska and inside the post.
The away goal left Celtic needing five to progress but one was looking unlikely as they struggled to break their visitors down.
Samaras stretched his leg towards a Juarez cross but could not turn it on target and Celtic almost fell further behind when they were exposed down their left side.
Matheus got away from Jos Hooiveld but Zaluska narrowed the angle and blocked at his near post.
Hooiveld was beaten for pace on another break and was booked for pulling back Paulo Cesar as Celtic limped towards half-time.
Mulgrew was taken off after struggling in the first period with Ledley moving to left-back and Marc-Antoine Fortune coming on.
Cesar played a poor ball across goal as Celtic were outnumbered on the counter-attack and the reprieve was soon followed by a 51st-minute leveller.
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.
The goal lifted the crowd and the team but they soon needed a saving tackle by Cha to prevent Leandro Salina converting after a back-post cross.
Maloney limped off in the 65th minute with former Sunderland forward Daryl Murphy coming on.
Celtic needed to follow up Hooper's goal quickly to put Braga under any sort of pressure but they failed to produce any sustained attacking sequences.
Samaras curled a 20-yard free-kick just wide and then headed a Juarez free-kick just beyond the post, but Zaluska was forced into a smart parry to deny Elderson's powerful shot from the left channel.
Celtic went ahead in the 79th minute when Juarez's deft header flashed in following an inswinging cross by Samaras.
They pushed Braga back for the remainder of the tie but could not threaten again as their Champions League campaign ended at the qualifying stage for the second season running.