Coyte and Erin Osborne (15 not out off 11) then finished the job with an unbroken partnership of 30 from just 3.4 overs.
After successfully chasing down 249 in game one at the SCG on Wednesday, the White Ferns smashed 288-6 after being sent in to bat.
Amy Satterthwaite (109 off 119 balls) posted her maiden ODI century in her 45th innings, just two days after being dismissed for 98, her seventh half century.
She struck nine fours and two sixes and combined with her captain Suzie Bates (59 off 86) for their second century stand of the series.
Satterthwaite and Bates, who scored an unbeaten 122 on Wednesday, inflicted more punishment on the bowlers following their 168-run union in game one. They put on 115 in just over 25 overs.
NZ advanced to 240-2, as Satterthwaite and Sara McGlashan (51 off 51) kept the pressure on Australia with a 102-run stand at just over a run a ball after Bates was run out.
Katie Perkins (31 off 15) ended the innings with a flourish, while Coyte (2-55 off 10) was Australia's only multiple wicket-taker.
The series continues on Monday at North Sydney.
- AAP