Rennes took the lead after 21 minutes when Jires Kembo (pictured) took a free-kick from the left and bent the ball into the net.
The home side seemed set for an easy win when Julien Feret and Yacine Brahimi both netted from outside the box in the 54th and 68th minutes.
However, Evian hit back with a Kevin Berigaud penalty and a Sidney Govou goal five minutes later. They could have forced the match into extra time but a Cedric Barbosa shot struck the woodwork with two minutes to go.
Three other games scheduled for Tuesday were postponed due to the freezing conditions.
The tie between amateurs Bourg-Peronnas and Marseille and the Ligue 1 clash between Dijon and Paris Saint-Germain will take place on February 15. Orleans will travel to Quevilly on February 21 for a fourth-tier clash.
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