Troyes' wait for a first victory of the Ligue 1 season continues after Toifilou Maoulida claimed an 88th-minute winner for Bastia at Stade Armand Cesari following two red cards for the visiting team.
Rock bottom Troyes, who have just two points from eight games, had hope at 2-2 as Marcos dos Santos twice equalised following Rincon's own-goal and Anthony Modeste's strike.
But Fabrice N'Sakala received a second yellow card in the 52nd minute and Thiago Xavier was sent off in the 68th before Maoulida eventually made the numerical advantage count for his side.
The opening goal came in the 16th minute after a poor clearance by Maxime Colin, who fed the ball directly to Sambou Yatabare. He fired the ball across the area where it struck Rincon and ricocheted over the line.
Dos Santos levelled nine minutes later when goalkeeper Macedo Novaes parried Thiago's headed effort straight into his path.
Jean-Christophe Bahebeck went close for Troyes soon after but this time Novaes' stop did the job.
Bastia had the lead again in the 37th minute when Wahbri Khazri got away from his man and fed Modeste, who produced a neat finish with his instep.
Troyes hit back four minutes after the interval when Dos Santos converted from the penalty spot.
Bahebeck won the decision but replays suggested there was minimal, if any, contact from Jeremy Choplin.
N'Sakala got his marching orders just three minutes later and he was followed quarter of an hour later by Xavier.
He can consider himself somewhat unfortunate - sent off for dissent against the referee after he wrongly awarded a penalty for handball.
Modeste missed the spot-kick, but with Troyes down to nine men it seemed unfeasible they could hang on.
Bastia did not dominate in quite they way they would have expected to for the next 20 minutes, but they did enough to take the points when Maoulida met Gael Angoula's cross and beat Yohann Thuram from 12 yards.