Cluj, who secured their first points at home in Group H after playing without a genuine centre-forward, now have seven points, the same as Galatasaray who beat leaders Manchester United 1-0 in Istanbul.
The Romanians visit United in their final group match on December 5, knowing only a win combined with a draw or a loss for Galatasaray at Braga would secure second place for them.
Pedro, a 24-year-old Portuguese who had never scored in European competition before, opened the scoring on a cold night in Transylvania after seven minutes following a delightful assist by Rafael Bastos.
He made it 2-0 eight minutes later with a cool finish that gave former CFR Cluj keeper Beto no chance after lively Senegalese winger Modou Sougou's fine run on the right flank.
The unmarked Alan reduced the deficit immediately, scoring his fifth goal in the competition after taking advantage of a mistake by Ionut Rada but Braga's joy was short-lived.
Pedro hit the target for a third time in the 33rd minute with Sougou turning provider once again.
Braga's final away match in the group went from bad to worse in first-half stoppage time when Brazilian defender Douglao was given a straight red card after elbowing Rada.
Bastos missed a golden scoring chance early in the second half and defender Camora hit the post while Braga substitute Mossoro's goal was ruled out for offside.
