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 Zealandwomen's coach.
He opened the scoring after 16 minutes from the spot and, after Kieran Govers hit the post from a penalty corner, crossed for Matthew Butturini to tap in just before half-time.
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 Pietersebut 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.