The Chilean former Arsenal star scored both his goals in the first half on the Cote d'Azur, with on-loan Arsenal wing-back Nuno Tavares on target in between.
The win put Marseille provisionally on top of the early Ligue 1 table with 10 points from four games, ahead of Lens on goal difference.
However, reigning champions Paris Saint-Germain are just a point behind and can reclaim top spot when they host Monaco later on Sunday.
Sanchez made his first start for Marseille last weekend after signing a one-year contract with the former Champions League winners following his departure from Inter Milan.
Against Nice, he opened the scoring in the 10th minute, taking a touch in the penalty box and firing into the roof of the net past goalkeeper Kasper Schmeichel.
Portuguese Under-21 international Tavares made it 2-0 in the 37th minute, his low strike from outside the area beating Schmeichel at the goalkeeper's near post.
It was his third goal in four games since arriving on loan from the Gunners, for whom he made 28 appearances last season after signing from Benfica.
Sanchez then followed in to make it 3-0 before the interval after a Tavares shot had been parried.
The result leaves ambitious Nice still without a league win this season.
Ivory Coast international Nicolas Pepe failed to make an impact for Nice on his debut after also signing on loan from Arsenal.
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