Following a midweek 3-0 routing at the hands of Porto in the Champions League, the French champions began the game cautiously and timidly.
Indeed, the hosts gave away possession clumsily in the ninth minute, which saw Stephane Sessegnon well placed to fire home following a cutback into the box, but the ex-Paris Saint-Germain man blasted wide.
Although the visitors kept a clean sheet against PSG last week, they were not able to repeat the feat against the principality side, with Radamel Falcao charging into the box to turn home Rony Lopes's low cross.
Montpellier toiled against their opponents and, save for a few half-chances, did not seem to have enough to trouble Monaco.
The result means that Monaco sit second in Ligue 1 on 19 points, level with PSG but with an inferior goal difference and having played a game more, while Montpellier climb into 10th place.