Both sides made a slow start to the game with the first chance coming only after 19 minutes, Diego Rolan watching his header from close range getting saved.
The hosts continued to dominate proceedings as the first half progressed, Rolan heading Gregory Sertic's corner just high and then having his shot saved.
The pattern resumed after the break, as Ludovic Obraniak was unlucky not to break the deadlock in the 54th minute when he shot against the bar from outside the box.
Henri Saivet tried to double his side's advantage a minute later, but his shot from outside the Bastia box was too high.
Diabate had an attempt saved midway through the second half and Obraniak came close in the final 15 minutes of the match, but Bordeaux had to settle for a 1-0 win.