Claudio Ranieri's side suffered defeat to Lille last time out, and are now two games without victory in their attempts to keep up with the other pacesetters at the summit of France's top division.
Daniel Wass gave Evian a surprise 20th-minute lead at the Stade Louis II, when he ran on to a hopeful punt forward and fired a half-volley into the bottom corner for his second goal in as many games.
Radamel Falcao controlled a through-pass from James Rodriguez before drilling inside the post from 12 yards to equalise for Monaco seven minutes later, but neither the £50m Colombian nor his teammates could breach the visiting defence again.
Evian themselves threatened on just as many occasions as their big-spending opponents and either side could have snatched maximum points in what was a very open finish to the contest.
A shot from the edge of the box from Dan Nistor forced Subasic to make another fine stop, which nearly proved doubly important when Monaco came close to getting a late winner.
Lucas Ocampos, who had earlier hit the side-netting, was denied by an even smaller margin on 88 minutes when his half volley from the edge of the box struck the crossbar.
The result leaves Monaco third in the table, level on points with second-place Lille, but two behind PSG. Meanwhile, Evian are now 12th having gone three games unbeaten.