Burl took a maiden five-wicket haul as Australia were bowled out for 141 - their lowest score in matches against Zimbabwe.
Australia were 129 for five when Burl started a three-over spell that returned five wickets - including the scalps of David Warner and Glenn Maxwell - for the cost of just 10 runs.
Warner was six runs short of his century when he was dismissed, but Maxwell (19) was the only other home batter to reach double figures.
Zimbabwe captain Regis Chakabva guided his side over the line with a measured 37 not out from 72 deliveries as his side won with three wickets and 11 overs to spare.
Australia's Mitchell Starc took the wicket of Burl to become the fastest man to reach 200 ODI wickets, but it did not prevent Zimbabwe claiming victory after being beaten in the first two matches of the series.
Zimbabwe coach Dave Houghton hopes the result will lead to more fixtures against some of the sport's powerhouses.
He told reporters: "I hope it goes a long way to providing that. The more wins we can get on in tours like this, the more that our chances are we might get some bigger opponents on a regular basis."
Houghton said the gulf between his side and major cricketing nations was still wide but Saturday's result would give belief to his players.
"If we'd lost the toss today we would have seen a game similar to the second game," he said.
"The gulf is huge - make no error - we would struggle to beat your state sides on a good day.
"We are growing and hopefully we grow quite quickly and with the amount of cricket we're starting to play and the fact that we are getting a few results like this...it's nice to get these little reward top ups to show the guys what they're doing is right."