A photo finish was needed to determine third place, with the 36-year-old edging out Hungary’s Kolos Csizmadia for bronze in a time of 35.202 seconds.
Heath had been narrowly beaten in the heats and the semi-final by the Hungarian.
Csizmadia’s compatriot Sandor Totka took gold in 35.035 seconds with Manfredi Rizza of Italy winning silver.
Heath has now won medals at three consecutive Games and set an Olympic-best time of 33.985 during qualifying.
Speaking after the race he said: "I was maybe a bit hesitant off the start and didn't reach my potential in terms of peak speed.
"I am still happy with the performance."
Asked how it felt to win another medal, he replied: “It is hard to put into words. It is what you work towards to be at your best for these events.”