Marseille were the early season pacesetters, but have now suffered three defeats in their past five league matches.
Genevois opened the scoring two minutes after the break at the Allianz Riviera when he tapped home Valentin Eysseric's drilled cross from the left after team-mate Carlos Eduardo somehow failed to make contact.
The visitors were handed a way back into the match when Kevin Gomis was sent off for a second bookable offence for a challenge on Dimitri Payet.
Marseille failed to capitalise on the numerical advantage, though, and they were punished for a defensive lapse when Hult finished from Alassane Plea's pass.
Florian Thauvin pulled one back in the 77th minute when he finished well after collecting Giannelli Imbula's pass over the top.
However, it was not enough to spark a late revival as Nice deservedly held on for three points.
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