Morgan Amalfitano headed home in the 33rd minute to lift Marseille to 18 points, four ahead of second-placed Lyon who drew 1-1 at Lille thanks to a late Lisandro Lopez equaliser.
Lille, who won the title in 2011, are 12th on seven points having won only one of their opening six league games.
Marseille are the first team to win their six first league games since Monaco completed the feat at the start of the 1960-61 season. Monaco went on to be crowned champions at the end of that campaign.
Three days after a tightly-contested Europa League draw at Turkish side Fenerbahce, Marseille were visibly tired and struggled to create chances against 17th place Evian.
"It was a tough game, we paid for the effort we put it on Thursday," captain Steve Mandanda told French channel beIN SPORT.
"We won with our heads and our hearts. We have been showing a lot of solidarity since the season started. But there is still a long way to go in the league."
Marseille went ahead with their first clear-cut chance, when Amalfitano headed home a neat Mathieu Valbuena free-kick.
The midifielder then missed the chance to double the lead when he sent another header narrowly wide three minutes later.
The hosts were made to work hard after the break as Evian laid siege to Mandanda's goal but an impressive rearguard display from defender Nicolas Nkoulou helped keep the visitors at bay.
France's reserve goalkeeper did well to save a powerful Jonathan Mensah header with nine minutes remaining to maintain Marseille's perfect start to the campaign.
Lyon are now four points adrift at the top of the table after drawing with Lille who were hoping to bounce back after their shock 3-1 home loss to Belarus's BATE Borisov in their Champions League opener on Wednesday.
Frontman Nolan Roux put Lille in front in the seventh minute when he poked the ball in from close range after Ryan Mendes had miscued a Mathieu Debuchy pass.
Lille dominated but lacked the cutting edge to double their lead.
Lyon's keeper Remy Vercoutre saved a Mendes shot on the stroke of halftime and superbly parried Dimitri Payet's neat back-heel on 57 minutes.
Lisandro eventually levelled for Lyon with ten minutes to play with a stunning goal from 25 metres to frustrate Lille, who also wasted a one-goal lead in their last league game at bottom club Troyes.
Bordeaux drew for the fourth consecutive time in Ligue 1 after a stoppage-time goal from substitute Chahir Belghazouani secured Ajaccio a point in a 2-2 draw.
Belghazouani chested the ball down to strike the last-gasp equaliser past goalkeeper Cedric Carrasso.
Bordeaux, who have not lost a league game since early April, are sixth on 10 points from six games. Ajaccio climb up to 13th on six points.
Laborious Bordeaux did not create a single chance in the first half, three days after thrashing Club Bruges 4-0 in their Europa League group stage opener.
Centre back Henrique and forward Yoan Gouffran both headed home after the break to put hosts Bordeaux in front, but Ajaccio twice recovered thanks to Ricardo Faty and Belghazouani.