Leonardo Jardim's men secured victory after scoring three times in the space of only 14 minutes in the second half.
Rony Lopes struck first in the 69th minute with a header, followed by counter attacking former Manchester City forward Stevan Jovetic. Radamel Falcao added the gloss after being introduced as a substitute.
Meanwhile Emiliano Sala managed an added time consolation for Nantes, who are now managed by Miguel Cardoso.
Monaco came second in the race to win France's top league last season - missing out to eventual champions Paris Saint-Germain (PSG).
PSG play host to Caen on Sunday at the Parc des Princes.
Thomas Tuchel will be without Italy midfielder Marco Verratti following close season surgery.
"He had an operation at the end of last season and he had an up-and-down pre-season," said Tuchel, who replaced Unai Emery this summer.
"Marco has trained very hard but felt a twinge at times so we have to be careful. But there is nothing worrying."
While Verratti will watch from the stand in the capital, one Italian is set to be on the pitch. Gianluigi Buffon joined his compatriot at PSG this summer, and the iconic former Parma and Juventus goalkeeper has already had a significant impact on the squad.
"It is easy with him as he is very professional - a legend.
"He is very humble and very professional - he has a lot of influence on our team," explained Tuchel, who said he is yet to decide if Buffon will be his first choice goalkeeper this season or one of Alphonese Areola or Kevin Trapp.
"They all have the talent to be number one!"