Saints, who had lost just once in the league prior to Sunday's trip which pitted top against bottom, forced the additional time thanks to Regan Grace's dramatic last-gasp try.
But man of the match Smith showed great composure to land the winning kick when the Broncos took a couple of tackles after a knock-on from Saints to set up the winning moment.
There was no sign of what was to come when tries from skipper James Roby and Luke Thompson helped Saints to a 12-0 lead after 10 minutes.