Arsenal's hopes of qualifying for the Champions League knockout stage suffered a blow as Matheus scored twice late on to earn Braga a famous victory.
The striker latched on to a long ball with seven minutes left to clip past Lukasz Fabianski, before running through in injury time to smash home.
Arsenal should have had a penalty at 0-0, but Carlos Vela was booked for diving after appearing to be fouled.
The Gunners also saw captain Cesc Fabregas limp off in the second half.
The late goals completed another miserable day at the office for manager Arsene Wenger and his team, who went into the game desperate for the point that would secure their last 16 place.
But after the shock of throwing away a two-goal lead to lose in the Premier League at home to arch rivals Tottenham on Saturday, another dramatic collapse was the last thing they would have wanted.
And now they must beat Group H whipping boys Partizan Belgrade at the Emirates on Wednesday, 8 December to progress, knowing that if they slip up, Braga could come back to haunt them once more.
They might have to do it without Fabregas, who limped off halfway through the second half with what looked like a recurrence of his hamstring problems, making Wenger's decision to play him when he rested the likes of Andrey Arshavin and Robin van Persie seem a strange one.
Having said that, an 11th successive qualification for the Champions League knockout stage appeared to be well within Arsenal's grasp for most of the game as they held Braga - who they beat 6-0 in September - at arm's length at the Estadio Municipal.
For all their willing, the Portuguese side simply did not look like having the ammunition to hurt the visitors, the Gunners comfortably dealing with any pressure they were put under without ever committing too many players forward.
Braga were restricted to pot-shots from the edge of the box as Matheus and Luis Aguiar both drilled wide, with Johan Djourou and Sebastien Squillaci doing well to keep Fabianski out of the game.
Arsenal burst into life sporadically, with Theo Walcott foiled by goalkeeper Felipe and Fabregas forcing the keeper to touch over his free-kick, but the game came to life when the Gunners' captain hobbled off with a hamstring problem that will keep him out of the game against Aston Villa on Saturday.
His replacement Samir Nasri injected some life into the visitors' display and they thought they had the perfect opportunity to take the lead when Nasri's fellow substitute Vela was taken out by Alberto Rodriguez inside the Braga area.
When Hungarian referee Viktor Kassai pulled out a yellow card for Vela instead of pointing to the spot, the Arsenal players reacted with astonishment - and within six minutes, they were behind.
With Emmanuel Eboue off the pitch receiving treatment for an injury, Matheus raced on to a ball from deep inside the Braga half from Brandao and kept his cool to slip the ball past Fabianski.
Arsenal tried to get back into it but they were hit by another sucker-punch deep into stoppage time when Matheus collected the ball just inside the visitors' half, jinked his way into the box and thrashed a thunderous left-foot finish in off the crossbar to wrap it up.
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