Trailing by three points with 1 1/2 minutes left, Smith led the 49ers on a seven-play, 80-yard drive, with the key play a 47-yard catch-and-run by Davis.
Smith's 28-yard touchdown run only 2 minutes earlier had given the 49ers a 29-24 lead, but Drew Brees came right back with a 66-yard TD pass to Jimmy Graham followed by a successful 2-point conversion.
Smith finished 24-of-42 through the air for 299 yards and three passing touchdowns for San Francisco, which advanced to the NFC championship game against either Green Bay or the New York Giants.
Vernon finished with seven grabs for 180 yards and a pair of scores, while Michael Crabtree added four receptions and a touchdown.
Brees set a Saints playoff record with 61 pass attempts, completing 40 for 462 yards, four touchdowns and two interceptions.
Darren Sproles established an NFL mark for most catches in a post-season game with 15 for 120 yards and a touchdown, but also lost a fumble -- one of five turnovers in New Orleans' loss.