Monaco, second in the table, are now 13 points adrift of leaders Paris St Germain with seven matches left to play.
PSG, who host Chelsea in the first leg of the Champions League quarter-finals on Wednesday, won 1-0 at Nice on Friday thanks to an own goal from Thimothee Kolodziejczak.
Evian are fourth from bottom on 34 points, five points clear of the relegation zone.
The home team, in a good run of form having lost just one of their previous four games, went close through Facundo Bertoglio and Clarck Nsikulu in the first half.
Bulgarian striker Dimitar Berbatov spurned Monaco's best scoring opportunity in the opening period.
The visitors also wasted an excellent chance early in the second period when Valere Germain, on as a substitute for Lucas Ocampos, set up James Rodriguez who missed with the goal gaping.
Mongongu eventually broke the deadlock with six minutes to go and Evian could have stretched their lead when a late free kick from Daniel Wass was saved by Croatian keeper Danijel Subasic.
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