The Walton-on-Thames doctor got off to a slow start but stormed back in the final 200m to take third place behind Australia's Ken Wallace and Canada's Adam van Koeverden.
The 31-year-old, already world and European 1,000m champion, won his gold on Friday and admitted after that race: "I knew I'd win from the start."
Brabants, who won bronze in Sydney and came fifth in Athens in 2004, said: “That was exactly the race plan we wanted, that is what we have been working on all year how to race an Olympic final.
"In the first two strokes there was no doubt I was going to win the race. No-one was going to come past me, I felt fantastic.
“I know it is easy to say when I won but right from the start line I was going to win the race.
“Four years ago I had unfinished business. I was good four years ago but I wasn’t this good.
“The guys at the last Olympics deserved to get those medals.”

Brabants revealed he has been waiting 15 years for this crowning glory of his career.
He said: “It doesn’t really feel real but that was what we have been working towards.
“The times when you are absolutely falling to pieces in training this is what it is for and what it has resulted in.”
Brabants won from defending Olympic champion Eirik Larsen, who came on strong at the finish to finish second.