The five-time world player of the year took his tally to 179 in 279 matches and although they all came from set-pieces - two penalty strokes and a short corner - the 33-year-old's influence in the overall performance was significant.
Dwyer went level with Mark Hager, who is at these Olympics as New Zealand women's coach.
After the break Chris Ciriello converted a low penalty corner drag flick and within a couple of minutes Dwyer poked home from close range at a short corner.
Then, with only the goalkeeper to beat, he was brought down by Erasmus Pieterse but picked himself up to score from the spot for his hat-trick.
Glenn Turner's 62nd-minute finish was made all the more simple by the brilliant passing move which went before it - Dwyer involved again - as Australia settled for six.
The Kookaburras, who have won Hockey medals at the last five Games, had scored 10-plus goals against South Africa twice in the last four years.
They have lost just two matches at a major tournament in the last three years.
In the later game, Great Britain's men got their Olympic campaign off to a winning start with a 4-1 victory over Argentina.
Even Pedro Ibarra's late penalty corner could not give Argentina a way back into the game.
Elsewhere, The Netherlands recovered from squandering a two-goal lead to just edge out eight-time Olympic Games champions India 3-2 while Republic of Korea beat New Zealand 2-0, Spain drew with Pakistan and Germany defeated Belgium.