Bordeaux had the first chance of the match inside the opening two minutes when Ludovic Obraniak hit a left-footed shot from just outside the box on the right, but Mickael Landreau managed to make the save.
Obraniak had another go at goal 10 minutes later from a similar position, but this time his effort flew wide of the far post.
Bordeaux did manage to take the lead in the 33rd minute when Saivet curled a free kick from the left of the box into the top near corner.
The hosts continued to dominate, with the lively Obraniak forcing another fine stop from Landreau, who turned the midfielder's shot past the post.
Bordeaux maintained their control of the play in the second half, but failed to create many chances bar Nicolas Maurice-Belay, whose effort flew over the upright.
The win moves Bordeaux up to seventh, while Bastia slip to 13th.