Second-half goals by Eric Bautheac and Valentin Eysseric left PSG, who scored through Zlatan Ibrahimovic, third in the table with 26 points from 15 games.
France midfielder Blaise Matuidi, one of PSG's best players this season, was sent off in injury time.
An angry Ancelotti told reporters: "Five points behind Lyon. We need to find a solution quickly. There will be something radical. I'm very upset by the team right now."
Earlier, Bafetimbi Gomis notched his fourth goal in as many days to earn Lyon a 1-0 home victory over champions Montpellier.
Lyon lead arch-rivals St Etienne, the team they meet at Stade Geoffroy Guichard next weekend, by five points after Les Verts drew 0-0 at AC Ajaccio on Friday.
Inspired by in-form Clement Grenier, the leaders outclassed Montpellier in the first half and denied them any goal chances.
The 21-year-old playmaker laid on the only goal when he took possession in his own half before setting up Gomis who netted with a powerful effort.
Montpellier, who are down in 13th spot and had scored in their 12 previous away league matches, pushed hard for an equaliser but could not make the breakthrough.
PSG have found the net in their last 12 league games but that statistic was little consolation after Saturday's defeat.
The visitors went close to opening the scoring through Jeremy Menez and Ibrahimovic early in the second half but it was Nice who struck first when Bautheac netted from close range in the 76th minute.
Ligue 1's top marksman Ibrahimovic levelled seven minutes later with a powerful free kick, the Swedish international's 13th league goal of the season.
Nice had the last word though as Eysseric latched on to a Bautheac cross at the far post to beat keeper Nicolas Douchez four minutes from time.
Olympique Marseille, who also have 26 points, are fourth on goal difference ahead of Sunday's game at Stade Brest.
- Sports & Recreation
- Zlatan Ibrahimovic