Bastia were forced to defend for long spells as Nice dominated possession, but the visitors were agonisingly close to taking a lead into the break when Francois Kamano fired just over from the edge of the box.
The away side's preference to operate on the counter-attack resulted in them opening the scoring early in the second half as Ayite latched onto a long ball before slotting past Mouez Hassan.
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Despite falling behind, Nice remained on top at Allianz Riviera, and the stubborn Bastia defence were regularly threatened as the hosts went in search of an equaliser.
Nice's best chance fell to Didier Digard late on, but he could only head over from close range as Bastia secured just their second win of the season.