Monaco were gifted the opportunity to break the deadlock in just the second minute when Olivier Sorlin handled inside the box, allowing Moutinho to slot home from 12 yards.
Youssouf Sabaly attempted to produce a response for Evian moments later, but his speculative strike from over 30 yards out sailed wide of the target.
Despite their pleasing start, Monaco found it difficult to create further opportunities, and they wasted a half-chance when Lacina Traore fired wide from a tight angle.
However, the hosts improved in the latter stages of the first half as a clever free kick from Moutinho found Yannick Ferreira Carrasco, who struck his effort just past the post.
Evian should have levelled the scores before the interval, but Daniel Wass could only head wide of the target after finding space inside the box to meet a Fabien Camus cross.