Christophe Galtier's side opened 2014 with a third home win in a row, a result that lifts them above Bordeaux and seven points behind third-placed Lille.
The hosts dominated the first half, with Franck Tabanou going close with two long-distance strikes, and they got their reward in the 22nd minute.
Hamouma's ninth goal of the season proved crucial, as he benefited from some sloppy Evian defending.
The goalscorer struck a tame effort from outside the penalty area, but it was spilled by Evian goalkeeper Bertrand Laquait and a defensive mix-up followed.
Jonathan Mensah failed to clear properly and as a result, Hamouma stole in to stroke the loose ball into the back of the net.
Benjamin Corgnet almost doubled their lead, but Laquait came off his line and did well to save the striker's dink, while Jonathan Brison dragged wide just before half-time.
Evian had two good chances after the break, as Brice Dja Djedje headed over, while a cross from substitute Cedric Barbosa caused some brief panic.
But it was Saint-Etienne who finished stronger - substitute Brandao almost scored a second with eight minutes left, but missed the target - as they claimed an expected home win.
Evian remain 16th in the table as a result of their defeat.
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