Robinson hit a four and six at the end of the innings to reach an unbeaten 106, as the home side scored 6-181 in windy and cold conditions on Wednesday at Mount Maunganui.
New Zealand were 3-6 in the second over and in dire straits after losing the toss, with Ben Dwarshuis on a hat-trick.
But Robinson and Daryl Mitchell steadied the innings with a 92-run stand, a record for the fourth wicket in New Zealand-Australia men's T20s.
Matt Short had Mitchell caught in the deep for 34 amid several dropped catches by the Australians.
Robinson hit six fours and five sixes in his 66-ball knock.
While Travis Head dropped two catches, he was also involved in a masterful run out late in the innings.
Marcus Stoinis stopped a boundary and threw the ball between his legs to Head, whose excellent throw caught Bevon Jacobs just short for 20.
The absence of Glenn Maxwell with a fractured arm is a significant blow to Australia, who recalled Short, Stoinis and Xavier Bartlett for the opener in the three-game series.
Maxwell was bowling in the nets on Tuesday when a solid drive from Mitch Owen caught him near the wrist. It is unclear when the star allrounder will return to playing.
Australian captain Mitch Marsh stuck to his preference of sending the opposition in, despite Tauranga's Bay Oval historically favouring the team that bats first.
Australia are playing at the venue for the first time.
Adam Zampa only conceded 27 runs from his four overs, while Dwarshuis took 2-40.
7°C