Arsenal, who had needed only a draw to qualify with a game to spare, lost the Group H lead to Shakhtar Donetsk, 3-0 winners over bottom team Partizan Belgrade.
Brazilian forward Matheus struck two late goals to give Braga a 2-O win over Arsenal in the Champions League on Tuesday and deny the London team the satisfaction of sealing their progress into the second round.
Arsenal, who had needed only a draw to qualify with a game to spare, lost the Group H lead to Shakhtar Donetsk, 3-0 winners over bottom team Partizan Belgrade.
Braga drew level with Arsenal on nine points, but need to beat Shakhtar by a margin of four goals away on Dec 8, or draw and hope that Partizan take points off Arsenal at the Emirates.
"We have to believe. We could reach the end of the group with three teams on 12 points. But we know perfectly well that it will not be easy," Braga coach Domingos Paciencia said.
Braga had conceded six goals in their group-stage debut at Arsenal in September but were much improved at home, giving the English side little room to deploy their smooth passing game.
"We have to win the next match. With these results, Shakhtar becomes favourite to win the group," Arsenal coach Arsene Wenger told Portuguese television.
Substitute Elton created Matheus's first goal moments after coming on by supplying a delightful pass from his own half.
Matheus ran into the box and put the ball over Arsenal keeper Lukasz Fabianski on 83 minutes.
Matheus's second came deep into injury time, also from a counter-attack, with the forward surrounded by three Arsenal defenders but still managing to fire in a shot that went in after hitting the bar.
Few chances
"It won't be easy against Shakhtar, but in football anything can happen. We scored four goals in Sevilla and, who knows, maybe we can do it again," the striker said, referring to Braga's 4-3 win in Spain in the playoff stage.
In a largely uneventful and low-tempo first half, clear chances were rare for either side.
The Londoners' best opportunity came when Cesc Fabregas led a counter-attack and sent Theo Walcott racing through into the box on 35 minutes, but keeper Filipe rushed out to block the forward's shot.
Braga's finest effort came moments later, with Brazilian striker Lima sending a powerful drive from 25 metres just wide of Fabianski's right post.
Walcott posed a danger again five minutes after halftime with a free kick that went just over the Portuguese bar.
Failing to penetrate the Arsenal box, Braga opted for long-distance efforts, but Lima's diagonal shot on 58 minutes and a powerful drive by Luis Aguiar two minutes later both went just wide.
Arsenal coach Arsene Wenger then substituted his three forwards and Carlos Vela was unlucky not to be awarded a penalty when he was fouled by Alberto Rodriguez in the 78th minute.
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