Former Spain boss Moreno replaced Leonardo Jardim last week and began his tenure with a dramatic 2-1 triumph in the last-64 encounter at the Stade Louis II.
Keita headed home a cross from Wissam Ben Yedder after a goalless first half but Boulaye Dia restored parity in the 69th minute.
Reims were awarded a penalty when Adrien Silva fouled Hassane Kamara with six minutes left on the clock, but Lecomte kept out Moussa Doumbia from the spot.
Keita made the most of the reprieve, controlling Aleksandr Golovin's corner and rifling a crisp finish into the bottom-right corner with practically the last kick of the game.
Monaco return to Ligue 1 action a week on Sunday when they travel to Paris Saint-Germain, who visit the principality three days later.