Forward Olivier Giroud netted the winner midway through the second period after some superb control to stun last season's double winners, who opened their campaign with a 1-1 draw at Nancy last weekend.
Montpellier were well organised throughout and could be decent bets for the Europa League places this season judging by their display.
Last term's runners-up, Olympique Marseille, also struggled when they were held to a 2-2 draw for the second straight Ligue 1 weekend after Auxerre came from two down to share the points.
The visitors raced into the lead thanks to Loic Remy after three minutes and Andre Ayew just before the break but Auxerre, strugglers for much of last season, hit back through Alain Traore and Roy Contout in the second half.
"To lead 2-0 and come away with a draw is infuriating," Marseille coach Didier Deschamps told reporters. "I think we stayed in the dressing room [in the second half]."
Marseille, the 2010 champions, drew 2-2 at home to Sochaux on last weekend's opening day.
Promoted Evian picked up their first ever win in the top flight with a 1-0 home success over Nice in Sunday's other game.
Paris Saint-Germain and Olympique Lyon drew their matches on Saturday as France's big teams labour in the opening stages of the campaign.
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