The in-form visitors, who have won their last two Ligue 1 matches, took a 20th-minute lead through Daniel Wass, who latched on to the end of a long clearance before firing a half-volley into the bottom corner.
Monaco had previously threatened on several occasions, with Yannick Ferreira Carrasco central to most of their attacking play.
The Belgian midfielder's cross on 11 minutes just evaded Falcao in the middle but continued its way towards goal and forced Jesper Hansen into a smart save.
Hansen was called into action again shortly afterwards, when Ferreira Carrasco picked the ball up on the edge of the box and volleyed into the ground, forcing the Dane to tip over his own crossbar.
Evian then scored through Wass against the run of play, but they were ahead for only seven minutes, as Falcao netted Monaco's equaliser by drilling a 12-yard shot just inside the post, following a pass from James Rodriguez.