Leonardo Jardim's side, who were looking for a third consecutive league victory, started well with Elderson Echiejile's first-half opener but Benjamin Moukandjo equalised nine minuets from time.
Monaco could have moved level on 20 points with third-placed Olympique Lyon but the draw left them in seventh place with 18 points from 12 games, seven points behind leaders Olympique Marseille who host RC Lens on Sunday.
Echiejile put the home side ahead after 20 minuets when he fired home from Bernardo Silva's cross.
Monaco pushed for a second goal but forward Anthony Martial missed two clear chances before the interval.
Reims, who are 12th on 15 points, scored with their first shot on target.
At the end of a sharp counter-attack, substitute Nicolas de Preville crossed for Moukandjo, who coolly slotted the ball past Danijel Subasic to salvage a point for the visitors.
Champions PSG will narrow the gap with Marseille to one point if they beat Lorient away on Saturday.
(Reporting by Julien Pretot; Editing by Martyn Herman)
