Jacob Oram top-scored with 61 as the Black Caps were all out for 152.
Tuffey has played 24 Test matches for the Black Caps, taking 74 wickets at an average of 30.79.
Rush won eight All Black caps and captained the Auckland Blues to the Super 12 title in 2003.
The Black Caps had looked on course for victory with the destructive Brendon McCullum at the crease in the penultimate over.
Daniel Vettori made 44 as the Black Caps' final four wickets added 195 runs, with Shoaib Akhtar taking 5-48.
Australia's 17-run win over the Black Caps ensured the inaugural Chappell-Hadlee series will be decided in Brisbane on Friday.
BBC: SPORT | Cricket | Other International | Ponting breathes sigh of relief
Twelve runs came off the second over, bowled by James Franklin, and the Black Caps were under serious pressure.
Pakistan beat the Black Caps 1-0 in the two-Test series but were humbled 4-1 in the one-day series that followed.
The Black Caps dismissed the tourists for 154 runs before tea on the first day of the first Test in Christchurch.
The Black Caps were in disarray when Fleming took over for the third Test against England 10 years ago in Christchurch.
The Black Caps, fresh off their victory over England in the NatWest series, should provide a much sterner test than short-handed Ireland.
He was expected to take the new ball with Matthew Hoggard against the Black Caps at Seddon Park on Tuesday, 4 March.
Harris, who used a runner. was yorked for four by Glenn MgGrath, bringing a tremendous late effort from The Black Caps to an end.
BBC: SPORT | Cricket | Other International | Ponting breathes sigh of relief
The Black Caps went into the series already depleted with Jesse Ryder, Grant Elliott and Daryl Tuffey unable to tour due to various injuries.
"Test cricket for a lot of guys around the world is the most important, " he said, following the Black Caps' 2-0 Test series defeat against England.
The New Zealander's optimism is fuelled by the signing of fellow countryman Kane Williamson who has just helped the Black Caps to a World Cup semi-final.
Johnson made light work of the Black Caps batting order after Simon Katich had scored an unbeaten century on Saturday to put the hosts in control.
His stabilising presence will be paramount to The Black Caps' fortunes in this World Cup, and especially when they take on traditional foes Australia on Tuesday.
The hosts were frustrated by Vettori and Mills, yet they produced a disciplined effort with the ball and limited the Black Caps to just 52 pre-lunch runs.
It was a dream which perhaps first formed itself when Fleming led the Black Caps to victory over India in the 2000 ICC Knockout final in Kenya.
He had been in impressive form with the ball for Lancashire and would have returned to the squad for the first Test against the Black Caps at Lord's.
Scott Styris hit 55 as the Black Caps, who went into the game trailing 3-1 in the best-of-five series, recovered from 32-3 to set a target of 242.
With boundaries few and far between, the Black Caps were forced to trades in ones and twos as Afridi and Gul produced spells which were tailor-made for Twenty20 cricket.
The Black Caps made a solid start and had looked untroubled until opener BJ Watling was cleaned bowled by a superb Johnson delivery for 19 in the 14th over.
But Williamson (1-2) later found himself among the New Zealand wicket-takers as the Black Caps polished off an overwhelming victory which included six slip catches, Taylor pouching four of them.
However, Flynn and debutant Tim McIntosh had to rebuild after the Black Caps lost Jamie How early on when he was caught by Shivnarine Chanderpaul at point off Daren Powell.
With swing and seam on offer, the Black Caps had a good start to leave South Africa on 20-2 after the early exchanges, and perhaps regretting their choice to bat first.
Vettori, who will join Stephen Fleming in triple figures, will lead a Black Caps bowling attack that looks certain to include Franklin after the team management assessed the conditions at Hamilton.
But a four from Iain O'Brien (19) ensured the Black Caps avoided the follow on total of 180 as he and Chris Martin (four not out) added 16 for the final wicket.
The former Black Caps skipper escaped when Ambrose failed to collect an outside edge off Paul Collingwood's first ball but Broad ended his stay at the crease during a fine spell after lunch.
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