Italian striker Balotelli picked up a straight red card when he argued with referee Tony Chapron in the 68th minute but Nice held firm to secure victory thanks to Cyprien's 15th-minute effort.
Nice have 56 points from 26 games, three behind leaders Monaco who were held to a 1-1 draw at Bastia on Friday.
French champions Paris St Germain will reclaim second place and move within one point of Monaco if they beat Toulouse at the Parc des Princes on Sunday.
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Cyprien put Nice ahead with a fine right-footed effort from the visitors' first clear chance.
The hosts went close 10 minutes into the second half but Nice keeper Yoan Cardinale denied Zargo Toure and Francois Bellugou in quick succession.
The defeat left Lorient with 22 points, five from safety.
(Reporting by Julien Pretot; Editing by Toby Davis)
