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.
— London Broncos (@LondonBroncosRL) June 9, 2019
At that stage it looked as though a routine win was on the cards, but the Broncos were level at the break thanks to an interception score from Ryan Morgan and James Cunningham's converted try.
Mark Percival crossed to put Saints back ahead early in the second half but Morgan's second and a try from captain Jay Pitts had the Broncos in dreamland.
Grace appeared to have ruined the party with his try in the final minute, only for Smith to spark huge celebrations from the Broncos faithful and end Saints' seven-match winning Super League run.
There was more late drama in Sunday's other contest as Tony Smith's first game in charge of struggling Hull KR ended in a 19-18 reverse at home to Wigan Warriors.
It appeared as though the match was also headed for golden point until Sam Powell nailed a timely drop goal for the Warriors, who scored tries through George Williams, Dan Sarginson, Joe Burgess and Liam Marshall to ruin the day for former Leeds Rhinos and Warrington Wolves boss Smith.
Photo Getty Images. Caption: Morgan Smith of London Broncos