The Kookaburras needed to win by four goals or more, following a surprise 3-0 defeat in the first game of the series and a 3-3 draw in the second.
The Australians played with a spark missing in the first two games, and goals to Will Ockenden, Kieran Govers and skipper Jamie Dwyer had them 3-0 up just 16 minutes in.
Striker Glenn Turner led the way in the second half with three goals, before Black Sticks skipper Phil Burrows finally put the Kiwis on the board from a dubious penalty stroke in the 65th minute.
Kookaburras coach Ric Charlesworth said the Kiwis were a better side than their world ranking of seventh indicated.
"I think our opponents play pretty tough, hard hockey, and you have to match that," Charles told reporters.
"And maybe we were surprised, but we didn't take our chances back."
Charlesworth said he was pleased with the improving form of four-time world player of the year Dwyer, who spent more than four months on the sidelines with a knee injury.
"I think Jamie's getting there, he's not back to his best, but he's fit and getting some hockey under his belt," Charlesworth said.
"He's wonderful player at his best, and we had to be cautious with him."