The home side went ahead as early as the sixth minute, thanks to a left-footed shot from Florian Thauvin from the left side of the box.
Just moments later Bastia made it 2-0, as Thauvin fired in from close range to score his second goal of the evening.
Yoan Gouffran halved the deficit on the half-hour mark when he found the target from the centre of the Bastia box.
Bastia restored their two-goal advantage in the 56th minute when Wahbi Khazri scored from a free kick.
Ludovic Obraniak hit the Bastia post for Bordeaux from a direct free kick on 75 minutes, but that was the closest the visitors got in finding their second goal of the evening.