St Etienne, also in the race for a place in the European competition, earlier slumped to their first league loss of the year, 3-1 at Lorient, to stay four points adrift of third spot.
Les Verts, who have qualified for the Europa League preliminary rounds by winning the League Cup, could be out of the race if third-placed Olympique Lyon beat leaders Paris St Germain in the late kickoff.
PSG have a four-point lead over second-placed Olympique Marseille and will be crowned champions if they win.
St Etienne are sixth on 59 points, four behind Lyon and one behind Lille in fourth and Nice in fifth.
Evian, who booked their place in the French Cup final earlier this week, climbed up to 17th, two points above the relegation zone, thanks to their comprehensive home victory.
It left bottom-club Brest, who suffered their ninth loss in a row at home against Sochaux on Saturday, an unassailable eight points adrift from safety.
Evian opened the scoring in the 23rd minute when Kevin Berigaud superbly controlled the ball with his back turned towards the goal and fired past Nice goalkeeper David Ospina.
A Renato Civelli header hit the crossbar for Nice three minutes later in an entertaining game.
Mohamed Rabiu doubled the hosts's lead three minutes into the second half with a powerful shot and Saber Khlifa made it 3-0 from close range before taking advantage of Ospina being out of position to net an outstanding 60-metre strike.
Lorient had been the only team to beat St Etienne so far in 2013 when they won a French Cup quarter-final last month, and they matched that feat in the league when Kevin Monnet-Paquet's header put them ahead six minutes after the break
Fabien Robert doubled their lead in the 67th minute with another header.
Centre back Kurt Zouma made it 2-1 with 10 minutes left following a corner kick but Lorient contained St Etienne's late surge and Yann Jouffre wrapped it up in added-time, bending in a free kick from 25-metres out.