Claudio Ranieri's men drew to within five points of Les Parisiens courtesy of a deflected strike from Mounir Obbadi with 20 minutes to go
The decisive moment arrived in the 70th minute when, following relentless pressure from the visitors, Mounir Obbadi's deflected 25-yard strike defied Nantes goalkeeper Remy Riou.
Despite their victory, Monaco remain third in the table, a point behind Lille, who won at Toulouse by the same scoreline.
PSG's 3-0 win at Reims on Saturday gave the capital club a seven-point lead at the summit, but second-placed Lille trimmed back that advantage as they kept a ninth consecutive league clean sheet at home to Toulouse.
Toulouse were in the ascendency for long periods during the first hour, with Lille goalkeeper Vincent Enyeama to the fore, but the game turned when the visitors' Argentine forward Oscar Trejo collected his second yellow card of the second period for a tackle on Idrissa Gueye in the 57th minute.
Lille finally broke the deadlock five minutes from time when Pape Souare powered home a header from a Florent Balmont cross.
Elsewhere, Saint-Etienne moved into sixth, one place behind Nantes, courtesy of another 1-0 scoreline at Nice.
Mevlut Erding poached the only goal in the 20th minute but the game was marred before kick-off when the visiting supporters were ejected from the stadium after tearing out seats and throwing them towards home fans.