With Nice struggling at the wrong end of the table, Marseille had entered the encounter at the Stade Velodrome as favourites, but Eysseric settled the encounter in favour of the visitors with his well-measured set-piece in the 66th minute.
Claude Puel's men could have taken the lead as early as 12 seconds, Dario Cvitanich pounced on a loose pass before flashing a shot over the crossbar from the edge of the box.
The hosts then had a shout for a penalty when Gregoire Puel appeared to handball in the area when sliding to make a tackle.
Marseille then went close as the lively Florian Thauvin hit a half-volley that forced David Ospina into a smart near-post save.
Saber Khelifa replaced Andre Ayew shortly before half-time, and the forward flicked just wide at the near post in the 49th minute.
Nice would make the home side pay for their profligacy in the 66th minute, as Eysseric neatly lifted the ball over the wall into the bottom left-hand corner past the despairing Steve Mandanda.
Thauvin almost orchestrated an equaliser, cutting in from the right to drag a left-foot shot wide but, ultimately, Marseille were unable to claw their way back into the match.
It marks the third time this season Nice have defeated Marseille, following a 1-0 win in the reverse Ligue 1 fixture, and a 5-4 triumph in the Coupe de France.
The win lifts Nice into 11th, though a raft of sides are yet to play this weekend, while Marseille's grasp on fifth is now under threat.
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