An early chance fell to Diabate when he collected the ball in the corner of the six-yard box, but his strike from a tight angle was beaten away by Geoffrey Jourdren.
The Malian forward took full advantage of his next opportunity in the 17th minute to hand Bordeaux the lead by tapping in Maxime Poundje's cross.
Bordeaux continued to look the more threatening of the two sides throughout the first half until a great chance was wasted by Montpellier, as Hilton headed Siaka Tiene's free kick wide after finding space just a few yards from goal.
Victor Montano created a great chance for the hosts to level the scores in the early stages of the second half with an inch-perfect pass to Anthony Mounier, who failed to find the target under pressure from a couple of defenders.
Emiliano Sala should have edged Bordeaux a step closer to victory just after the hour mark when he latched onto a loose ball inside the box, but the substitute sent his effort into the stands behind the goal.
However, the visitors managed to withstand the late pressure to ensure their campaign started with three points.