Maupay gave Nice the lead in the eighth minute, making no mistake with a shot from inside the six-yard box.
Bautheac ensured that Nice would go into the break with a two-goal cushion when he directed the ball into the bottom-right corner after Carlos Eduardo's through ball.
© Getty Images
The result looked to have been put beyond doubt in the 53rd minute as Bautheac scored his second goal of the game direct from a free kick, powering the ball into the top-left corner of the net.
Toulouse had other ideas though, and Wissam Ben Yedder pulled a goal back in the 79th minute.
Just two minutes later, Oscar Trejo's shot found the roof of the net to pull Toulouse back to within a goal of their opponents.
The Toulouse faithful thought that a miraculous comeback was on the cards, and Trejo saw his shot saved with seven minutes remaining.
Jean-Armel Kana-Biyik then shot wide from a good position as Nice held on for the final-day victory.