The victory levelled the three-match series at 1-1 with the final game in Hamilton on Monday. Australia have not lost a bilateral one-day series since India in October, 2013.
Marsh scored an unbeaten 69 and combined with John Hastings (48 not out) in an unbroken 86-run seventh-wicket partnership for the visitors as they overhauled New Zealand’s 281 for nine with 3.3 overs remaining.
David Warner top scored with 98 in Australia’s 283-6, and combined with the recalled Usman Khawaja (50) in a 122-run opening stand that had set the foundation for their victory.
Mitchell Santner had almost taken the game away from the visitors with his left-arm spin, breaking the opening partnership and then trapping Warner leg before when the aggressive opener was within sight of his sixth one-day century.
Santner then had Matthew Wade brilliantly caught in the deep by Adam Milne for two to reduce the visitors to 197 for six in the 33rd over and give the hosts a sniff of victory in a game they had not played well enough to win.
Too many overs were peppered with boundary balls as they struggled with their length on a wicket that had little help for the bowlers.
They had also not posted a big enough target as their innings suffered from fits and starts with all of the batsmen, apart from Henry Nicholls, laying solid foundations only to fall when they needed to build.
Kane Williamson top-scored with 60, though the hosts’ total was only pushed to a score considered defendable courtesy of a 61-run eighth-wicket stand between Santner (45 not out) and Milne (36).
The 24-year-old Santner, who only made his international debut last year, is quickly impressing as an automatic selection in all formats of the game with his fluid batting and handy left-arm spin.
Saturday’s match showcased his blossoming capabilities, with New Zealand slumping to 193 for six in the 37th over and in danger of being bowled out before their full allocation.
Santner, who was joined by the right-handed Milne 12 runs later, however kept the strike ticking over to put the bowlers off their rhythm, turned singles into twos with hard running and hit the bad ball to the boundary.
Opening bowler Josh Hazlewood, however, ended the partnership, and stopped New Zealand from edging closer to a more competitive 300, to finish with 3-61 from 10 overs.