Paul Stastny also scored for St. Louis, which has won four of five. The Blues rallied from a 2-0 deficit and have a point in their last 13 home games.
Tomas Plekanec and Paul Byron scored for Montreal, which completed a five-game trip 2-3.
Goalie Jake Allen made 28 saves to improve to 13-3-3.
Schwartz lifted a backhander past goalie Al Montoya with 1:22 left in overtime.
Stastny and Schwartz scored in a 2:51 span in the third period to tie it at 2.
