In a high-scoring match which produced 26 sixes, New Zealand chased down Australia's 346 for five with one wicket and just three balls to spare here on Tuesday.
McCullum struck the winning runs, fittingly another boundary, to end unbeaten on 86 off 89 balls while Craig McMillan engineered the victory with a career-high 117 off 96.
The result further demoralised world champions Australia, who have now lost five matches in a row just a month away from the World Cup.
When Hayden walked from the pitch unbeaten on 181 - the highest score by an Australian in one-day cricket -- the Australians were in sight of a consolation win in the Chappell-Hadlee Trophy series.
New Zealand, already 2-0 up, had been set the second highest one-day run chase to win.
Australia's confidence grew as New Zealand stumbled to 41 for 4 with the new glamour batsman Ross Taylor -- the hero of the second match in Auckland -- already back in the pavilion.
But out strode McMillan and the tide turned dramatically.
He joined Peter Fulton to put on 75 in 46 balls for the fifth wicket before Fulton fell for 51.
Enter Brendon McCullum, back to his rightful place down the order after a failed experiment as an opener, and between them the Macs blasted New Zealand into contention.
By the time Watson bowled McMillan, New Zealand had the scent of victory with nine overs left and only a further 66 runs required.
Australia came back into the reckoning as Mitchell Johnson winkled out James Franklin and Daryl Tuffey reducing New Zealand to their last two wickets with 44 runs required in five overs.
Mark Gillespie atoned for bowling New Zealand's most expensive 10-over spell in limited-overs cricket - 2-83 - with a quickfire 28.
When he departed, McCullum hit a six and four off the first three balls of the final over to complete the series win for New Zealand.
Despite the heroics of McMillan and McCullum, Hayden was the man-of-the-match for his outstanding innings as he cashed in on New Zealand's decision to lay off their two most experienced bowlers, Shane Bond and Daniel Vettori.
His 181 came from 166 balls and included 10 sixes -- an Australian record and one short of the world record of 11 sixes jointly held by Pakistani Shahid Afridi and Sri Lanka's Sanath Jayasuriya.
Shane Watson chimed in with 68, but none of the other Australian batsmen were in long enough to post scores of note.
Brad Haddin made 38 from 65 balls, while Mike Hussey and Cameron White both made 13 and Brad Hogg 12.
Adam Voges, in his first one-day international, was not out five.