The 18-year-old opened the scoring in the 13th minute with a fine finish, then won a penalty after a spectacular run, before doubling his personal tally nine minutes from time.
His magnificent showing came at the end of a week in which he helped Monaco knock Manchester City out of the UEFA Champions League in the last-16 and was handed his first call-up for France's senior side.
Monaco sit three points clear of Paris Saint-Germain while third-placed Nice are seven points adrift.
Leonardo Jardim opted to rest Bernardo Silva and Tiemoue Bakayoko, with Joao Moutinho and Nabil Dirar coming into the side, but the changes did not hold Monaco back and they went ahead when Mbappe beat goalkeeper Remy Vercoutre after a quick one-two with Valere Germain.
Monaco then felt they should have been awarded a penalty after Thomas Lemar went to the ground inside the area late in the first half, but referee Antony Gautier waved away the protests.
However, Gautier did point to the spot shortly after the restart.
Damien Da Silva brought down Mbappe after a stunning run from midfield and Fabinho made no mistake from 12 yards to double Monaco's lead.
Mbappe had the final word, though, as he netted his 19th goal of the season with a fine header from Moutinho's cross from the left, before going off to warm applause late on.