Last season's runners-up, who got off to a mediocre start this term, lie in 11th place with 14 points, eight adrift of leaders Olympique de Marseille before taking on Benfica in the Champions League on Wednesday.
OM host Toulouse on Sunday (1200 GMT).
Portugal's Moutinho scored his first league goal of the season when he converted a second-minute penalty after Olivier Sorlin's handball in the box.
Striker Lacina Traore missed a second when his attempt went just wide after he had dribbled past Evian keeper Jesper Hansen.
Evian were dangerous on the break and Daniel Wass came close to equalising shortly before halftime, only for his header to shave Danijel Subasic's post.
In the 66th minute, Djakaridja Kona's header smashed the bar for the visitors and shortly afterwards, Evian were punished for their lack of accuracy as it was Monaco who doubled their tally.
Hansen was lobbed by Ricardo Carvalho's long clearance and Yannick Ferreira Carrasco only had to poke the ball into the empty net as the home side wrapped up the win.
Elsewhere, Lille slumped to a surprise 2-1 home defeat against En Avant Guingamp.
On Friday, champions Paris St Germain beat RC Lens 3-1 in an away game played at the Stade de France, but will have to make do without Edinson Cavani in their next league game after the Uruguay striker was controversially sent off.
(Reporting by Julien Pretot; Editing by Mark Meadows)
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