Neesham needs to prove his fitness for possible inclusion for the final two ODIs in the five-match series against India but he lasted just five balls, scoring one run.
One-time New Zealand international Mitchell McClenaghan captured three for 32 for the Aces, while left-arm spinner Ronnie Hira chipped in with a Richie Benaud reminder, taking two for 22 from four overs.
The Firebirds slumped from 72-3 when Neesham was caught by Black Caps teammate Martin Guptill in the 11th over to 122 all out by the 19th.
Auckland's run chase started badly with Guptill falling to a leading edge off the first ball of the innings.
Mark Chapman stayed strong, unbeaten on 47 from 41 balls as the Aces got home comfortably with 14 deliveries left.
It was just the second win of Auckland's campaign and keeps them alive in the race for the top three final berths.
• Meanwhile, at the Melbourne Cricket Ground, India pulled a masterstroke with the introduction of Yuzvendra Chahal in the deciding one-day international against India.
The 28-year-old took six for 42, his best ODI figures, which equalled countryman Ajit Agarkar for the best bowling figures at the MCG in an ODI.
"It's always an amazing feeling any time you do well for India," Chahal said. "I'm playing this game after so long and this is my first match in Australia. Whenever you take five wickets, it's a great feeling."
Chahal tore apart the Australia top order, with Shaun Marsh, Usman Khawaja, Marcus Stoinis and Peter Handscomb all victims as Australia were bowled out for 230 without seeing out their 50 overs.
"Obviously when you contribute for your team and it's my second five-wicket haul, it's an amazing feeling," he said. "It's the first time my sister has come to see me play."
The wrist spinner bamboozled the Aussies throughout the game with key wickets.
Aussie skipper Aaron Finch continued his lean run, out LBW for just 14. It's the eighth time in nine innings Finch has been out LBW or bowled.
Finch has scored just 26 runs in the three matches of the series.
- with news.com.au