A goal kick from Hansen helped Evian score against the run of play. Evian striker Pape Amodou Sougou redirected it toward Wass, who held off a challenge from Layvin Kurzawa to shoot past Subasic.
Evian came close to doubling the lead in the 25th when Sougou skipped past Monaco midfielder Jeremy Toulalan but Subasic quickly came off his line to deny the Evian striker.
Monaco dominated possession but it equalized on a counterattack. Colombia international James Rodriguez slipped a through ball to Falcao, who scored with a diagonal shot.
Evian was dealt a blow in the 33rd when Barbosa picked up a shoulder injury and was replaced by Nicolas Benezet.
Rodriguez took the Evian defense by surprise in the 43rd by sending a corner outside the area for Toulalan, whose low volley sailed wide.
Monaco pushed for the winner in the second half by playing with two strikers, as Emmanuel Riviere replaced Ferreira Carrasco in the 68th.
But it was Evian which created the best chances. Sougou charged down a clearance from Monaco centerback Eric Abidal to be clean through on goal in the 65th, but Subasic stopped his effort with his right foot. Subasic also parried long-range strikes from Evian substitute Dan Nistor in the 74th and the 84th. The Monaco goalkeeper was lucky to see Sougou fire over the bar from 10 yards in the 82nd.
Despite Evian's pressure, Monaco nearly netted the winner in the 88th when Ocampos struck the post from 18 yards.
"We have a lot of regrets," Sougou said. "But we should not be ashamed of our performance. If we had converted our chances, we could have won this match."
Evian provisionally moved into 12th place.
On Saturday, it's: Paris Saint-Germain vs. Nice; Bastia vs. Rennes; Guingamp vs. Lille; Lorient vs. Reims; Toulouse vs. Ajaccio; and Valenciennes vs. Montpellier.
On Sunday, Saint-Etienne hosts Lyon, Nantes visits Bordeaux and Marseille faces Sochaux.