Briton Ratcliffe's Ineos had its takeover of the French club, coached by Patrick Vieira, confirmed on Monday.
They lost his opening fixture at the helm to Marseille on Wednesday in a game which had been suspended following homophobic chants from the Nice supporters.
Sunday's win moved Nice onto nine points after four games, the same as Rennes and Angers as well as champions PSG who top the table on goal difference.
Vieira's side started badly in Brittany as defender Gautier Lloris, the younger brother of Tottenham goalkeeper Hugo, scored an own goal from a corner after 25 minutes.
The visitors dominated the second half and Wylan Cyprien converted from the spot with less than half an hour to play.
Coly's winner, two minutes into added time came as he scrambled home a corner with his knee.
Later, Marseille overcame Saint-Etienne 1-0 after their own fans released a pre-match banner in the stands directed at the league's organisers who have mooted the idea of introducing a list of banned words in football stadiums in an attempt to combat homophobia and bad language.
Dario Benedetto, one of only two major summer signings by new coach Andre Villas-Boas, claimed his second goal in two games after 33 minutes with tidy build-up play which included an intricate flick-on from the Argentine.
Saint-Etienne, who were looking for their first triumph at the Velodrome since 1979, had numerous second-half chances with the best falling to midfielder Zaydou Youssouf, but Steve Mandanda, making his 400th league appearance for Marseille, saved low to his right.
Also on Sunday, last season's runners up Lille lost 2-0 at Reims.
The visitors, who were down to 10 men after Yusuf Yazici received a second yellow card, conceded a penalty for a foul on Moussa Doumbia, and after a delay for video review, the winger converted in the 73rd minute.
Remi Oudin's cross flew into the Lille net for the home side 17 minutes later.
Elsewhere, new arrival Islam Slimani scored twice but Monaco could only claim a 2-2 draw at Strasbourg due Adrien Thomasson 84th minute equaliser.