Evian leapt out of the relegation places as they inflicted a first home defeat on Reims since the opening day of the season.
Kevin Berigaud and Youssef Adnane scored in either half to fire the visitors, who had lost their past three league games, 2-0 ahead.
A Cedric Cambon own goal with 18 minutes left meant they had to sweat on the victory but they held on to move ahead of Sochaux, who host Lyon on Sunday.
Reims had won four of their past five on home soil, but while Franck Sigorino forced Bertrand Laquait into a smart early stop, it was Evian who enjoyed the better moments in the first 45 minutes.
Sidney Govou picked out Berigaud in the centre of the park and his 20-yarder drew Kossi Agassa into action for the first time.
The Reims goalkeeper had more work to do moments later to deny Berigaud one-on-one after he had been released on goal by Aldo Angoula.
Berigaud opened the scoring on 17 minutes, however, to give Evian a deserved lead. Govou tried his luck with an angled drive that deflected off Berigaud and wrong-foot Agassa at close quarters.
Reims tried to rally and Laquait had to be at his best to keep out Nicolas Fauvergue's flicked shot after a poor clearance.
Evian took control of the match seven minutes after half-time when Adnane put them 2-0 ahead. Govou sent him racing in on goal and he coolly opened up his body before side-footing past the exposed Agassa.
Berigaud was denied a second when he tried to lob Agassa, but the keeper was not to be fooled, before the hosts grabbed a lifeline on 72 minutes.
Odair Fortes fired in a cross that Cambon headed past his goalkeeper but an equaliser never came as Evian climbed out of the bottom three.