Said Albert: "I played the rabbit and they were the lions."
Brown, one of the best chasers in Ironman history, made some ground on the run leg but couldn't haul in Albert. Bozzone ended further back in third, while Bevan Docherty pulled out of the race just over halfway through the run leg.
It was an emotional victory for Albert, who celebrated wildly, doing a return lap up the victory chute as he danced to Tina Turner's Simply the Best.
"I will never, ever forget this day," said Albert, who finished runner-up to Docherty last year. "If I let all of the emotions loose right now, I would be crying like a baby. Beating these guys is what makes it so special."
Albert led out of the water, pursued closely by a seven-man bunch, including Docherty and Bozzone. Brown was just over two minutes further back.
It was a chilly morning and the group compressed on the bike leg, leading to the thrilling spectacle at the 80km mark of 10 riders separated by less than 10s.
Albert made his move and then put the hammer down during the final 30km on the cycle leg. Brown, Docherty and Bozzone trailed the European by around six-and-a-half minutes at the start of the run.
After dropping a faltering Docherty - who was vomiting on the side of the course - and Bozzone, Brown powered home on the run leg, at one stage cutting three minutes off Albert's lead. But the Estonian held on for a 4m 21s victory, finishing in 8h 17m 33s.
Meanwhile, American Meredith Kessler took out the women's event in imperious fashion, beating the course record by almost five minutes. She finished in 9h 08m 46s, smashing the previous mark (9h 14m 35s) set by Jo Lawn in 2010.
"I love it here," said Kessler. "During the run somebody yelled, 'We've adopted you now', and that is honestly how it feels when I race here."
Gina Crawford, looking to repeat her 2009 triumph, put up a determined fight - and set the third fastest time in Taupo history (just over a minute shy of Lawn's mark) - but lost sight of the American in the early stages of the cycle leg.