The two sides came into the clash occupying the division's last two spots, but it was the home side who took the opportunity to climb out of the drop zone.
Evian took the lead shortly before the hour thanks to a spectacular free-kick from Daniel Wass, and they made sure of victory nine minutes from time when Nicki Bille Nielsen struck his first goal for the club.
That secured back-to-back wins for Evian who moved up to 16th in the table while Guingamp remain bottom following a third successive league defeat.
At the opposite end of the table, Saint-Etienne thumped Lyon 3-0 to move up to fifth and within a point of their opponents.
After a small fire broke out in one of the refreshment stalls before the game at Stade Geoffroy-Guichard, the match was able to go ahead without any further incident.
Moustapha Sall broke the deadlock in the 18th minute, and Ricky van Wolfswinkel doubled the hosts' advantage five minutes prior to the interval.
Any Lyon hopes of a comeback were extinguished when Renaud Cohade netted a third midway through the second period, and the visitors' miserable day was summed up when Alexandre Lacazette missed a penalty 11 minutes from time.
It ended Lyon's 10-match unbeaten run, and leaves them in third place, seven points off the leaders Marseille.
Sandwiched between Lyon and Saint-Etienne in fourth place are Bordeaux, who defeated Lille 1-0 thanks to Cheick Diabate's winner shortly after the hour.
