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Evian responded well to falling behind, but it was Monaco who should have found the second goal midway through the first half when Yannick Ferreira Carrasco fired wide from Moutinho's free kick.
Despite struggling to create chances before the interval, Evian could have levelled with their first clear opportunity, as Daniel Wass headed just wide from a good position.
The visitors looked comfortable in a quiet spell early in the second half, but their struggles in the final third continued as Wass missed the target by a yard with a long-range effort.
Monaco operated on the counter-attack as they looked to protect their lead, and they were rewarded with just under 20 minutes remaining when Carrasco converted from Ricardo Carvalho's pass to secure them a first win in three.