Keeper Laquait deflected a Romain Hamouma header into his own net in the 77th minute as Les Verts leapfrogged arch rivals Olympique Lyon on goal difference with seven matches remaining.
St Etienne, who are unbeaten in 16 games in all competitions in 2013, dominated possession throughout but increased the pressure until Laquait was forced into his costly mistake after Hamouma's header from a Francois Clerc cross.
"But if we wanted to win we first needed not to concede any goal and we defended well. The most important thing was to keep our rivals at bay."
Evian have a tough task on their hands as they dropped to 17th on 31 points only one point above safety.
On Saturday, Paris St Germain restored a seven-point lead at the top over second-placed Olympique Marseille with a 2-0 win at Stade Rennes.
Later on Sunday (1900), sixth-placed Lille will continue their bid for a European place when they host Lorient.