Ellen Brown, a Green Party candidate, finished third and will not advance to the general election.
In a state that leans heavily Democratic, Chiang is the favorite to win in November, after having served two terms as controller.
Conlon has never held elected office and fell short in a previous campaign for treasurer in 2002.
The treasurer helps manage the state's finances. That includes selling bonds that fund major projects and serving on boards overseeing California's public pension systems.