The result kept Marseille fourth on 19 points from 10 games, 11 points behind runaway leaders and champions Paris Saint-Germain.
Nice dropped to 14th on 11 points, only two above the relegation zone, after striker Mario Balotelli missed a string of chances in the first half.
Balotelli was twice foiled by visiting goalkeeper Steve Mandanda and a last-gasp tackle by a Marseille defender also denied him a clear-cut finish before Sanson struck against the run of play at the other end.
A slick move down the right flank allowed Sanson time and space to pick his spot and unleash a stinging low shot into the bottom left corner of the net from 13 metres in the 42nd minute, capping Marseille's first purposeful move.
Nice pressed hard for an equaliser in the second half but Marseille's resolute defending confined the home side to just a few half-chances as their patchy form continued.
Elsewhere, high-flying Montpellier maintained their good start with a 2-0 home win over Bordeaux which kept them third.
Gaetan Laborde headed Montpellier into a 17th-minute lead and an Andy Delort penalty sealed the contest.
Sixth-placed St Etienne were held to a 1-1 home draw by 11th-placed Rennes.
A Wahbi Kazri penalty fired St Etienne ahead before Ismailla Sarr equalised and Hatem Ben Arfa missed a penalty for the visitors.