The result, on the back of two consecutive defeats in all competitions, left Monaco in third place with 26 points from 13 matches, two behind leaders Paris St Germain who will extend their lead to five points if they beat Nice on Saturday at the Parc des Princes (1600 GMT).
Evian took the lead in the 20th minute when Daniel Wass caught the Monaco defence snoozing, outspeeding Lucas Ocampos before firing a fine, rising shot past Danijel Subasic.
Monaco levelled seven minutes later when Falcao beat Jesper Hansen with a low, diagonal shot from inside the box after collecting a defence-splitting pass from James Rodriguez.
Evian sat back in the second half but came very close in the 66th minute when Pape Amodou Sogou sneaked past Ricardo Carvalho and Eric Abidal only for the Senegalese striker to be denied by Subasic.
Monaco, who had lost 2-0 at second-placed Lille last weekend after being eliminated 1-0 by Stade Reims in a League Cup last-32 match, had another chance in the closing stages but Ocampos's strike smashed Hansen's bar in the 88th.
(Reporting by Julien Pretot; Editing by Rex Gowar)
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