The Argentine netted his fifth league goal early in the second half to help Les Aiglons to victory.
Despite being in control for the majority of the first half, Nice could not find a way through and were denied in the 11th minute when Nemanja Pejcinovic forced a save from Guy N'Dy from inside the box.
Cvitanich had a chance just short of the half-hour mark when he got a shot away from inside the box, but N'Dy made a stunning stop to push the ball away from the the top corner.
However, Nice took the lead just three minutes into the second half when Christian Bruls swept a free kick into the far post from the left touchline where an unmarked Cvitanich nodded the ball into the back of the net.
Nice move up to fifth in the table following the victory, while Guingamp drop to 14th.