Missing several key players through injury or suspension, Bordeaux, who lost their Ligue 1 crown to Marseille last May, looked sluggish and hardly created a chance.
Montpellier deservedly took the points courtesy of defender Garry Bocaly, who headed home a free-kick by Chile midfielder Marco Estrada early in the second half.
Marseille had kicked off their title defence with a shock home defeat by promoted Caen on Saturday while Lyon were held to a goalless draw by Monaco.
Sunday's result meant that of the top six teams in last season's final standings, only Montpellier, who finished fifth, won on the opening weekend.
The victory provided Montpellier with some consolation three days after they failed to survive the qualifying round of the Europa League, falling to Hungarian side Gyor.
"It was important for us to win tonight after losing in Europe and to show our fans that our goal is to have another great season," Bocaly told French television channel Canal Plus.
Bordeaux, notably without injured goalkeeper Cedric Carrasso and suspended central defender Marc Planus, threatened on 13 minutes when left-back Benoit Tremoulinas hit the post with an angled shot from the edge of the box but showed little else.
Paris Saint-Germain set the early pace with a 3-1 win over once-great St Etienne in a match marred by crowd trouble on Saturday.
Nine PSG supporters were held for questioning after violence erupted before the game in the French capital, police said on Sunday.
The nine, who could face charges, were among supporters who threw flares and briefly clashed with police outside Parc des Princes before kick-off.
The supporters protested against an anti-violence plan by the club preventing supporters choosing their seats. More than 200 fans were rounded up by police. Most were released almost immediately.
Follow FFT.com on Twitter
Join FFT.com on Facebook
