The 22-year-old former France Under-21 international forward scored two goals in 31 appearances at Paris Saint-Germain after coming through the youth set-up at the Parc des Princes.
Augustin then moved to RB Leipzig for €13 million in summer 2017, scoring 20 goals and providing eight assists in 67 games for the German Bundesliga outfit.
However, he has now moved back to France and will ply his trade at the Stade Louis II for at least the 2019-20 season.
Monaco have the option to sign him next summer and Augustin is relishing the prospect of trying to score goals for Leonardo Jardim's side.
He said: "I am very happy to be involved with AS Monaco in this ambitious project. After two seasons in the Bundesliga, I am looking forward to a new start in my country, and in a Championship that I know well. This is a new challenge that I am keen to take up. I really want to show what I can bring to the group to help them achieve club goals."