
Defender Souleymane Diawara headed in three minutes into the second half to hand the Ligue 1 champions all three points.
The victory left Didier Deschamps’s side third in Group F with three points from as many games, six points adrift of Chelsea who beat second-placed Spartak Moscow 2-0 in Russia.
“It was not an easy game but we did our job to claim this victory,” Diawara, who otherwise produced a mediocre performance in defence, told reporters.
“We broke the deadlock on a set piece once again before struggling towards the end of the game.”
Marseille’s Andre-Pierre Gignac was then twice denied by keeper Martin Dubravka as the hosts put the pressure on at a windswept Stade Velodrome.
Tomas Majtan came close to an equaliser shortly after the hour when he outpaced Cesar Azpilicueta but keeper Steve Mandanda pulled off a super save to parry the Zilina striker’s effort from inside the area.
“We now have to go and get a good result in Moscow but it will not be easy, especially because of the (cold) weather in Russia,” said Diawara.