Vaultier edged Australia's Jarryd Hughes into second place, while Spain's Regino Hernandez claimed bronze.
The top three riders from each of the eight heats advanced to the quarter- and semi-finals, with six then contesting the final.
Vaultier, a five-time World Cup champion, looked comfortable in the early stages of the competition but was involved in a three-man pile-up in the semi-final and fell over.
The Frenchman, however, was first up and qualified in third place.
Sochi silver medallist Nikolay Olyunin was also involved in the incident but could not finish the race after suffering an injury.
There was yet more drama in the final as Americans Nick Baumgartner and Mick Dierdoff collided with Australian Alex Pullin, who did not finish.
Vaultier then crossed the line narrowly ahead of Hughes and Hernandez.
(Editing by Greg Stutchbury)
