Australia thumped the hosts by an innings and 52 runs at the Basin Reserve shortly after lunch on the fourth day to put themselves in prime position to usurp India at the head of the ICC test rankings.
The triumph means Smith's side can afford to draw the second test starting in Christchurch on Feb. 20 as any series victory, whether it be by a 1-0 or 2-0 margin, will take Australia past India.
"We want to be number one in the world in all three formats and I think this test match is a step in the right direction," Smith told reporters. "I said when I took over as captain that I wanted to win every series away from home.
"It's great to be one-nil up in the series against New Zealand and hopefully we can take a step up in Christchurch."
Smith felt his team had put in a complete performance against Brendon McCullum's side, exploiting the green wicket on the first day, then building a patient innings when they batted to ensure they wore down the home attack as conditions improved.
"It was a pretty good toss to win. Having said that, we had to put the ball in the right areas for extended periods and I think we did that," he said.
"We put the New Zealand batters under pressure and reaped the rewards from that... and with our batting, we were willing to bat for long periods and get through the tough periods."
REVERSE SWING
McCullum also praised Australia's bowlers for their efforts on the first morning and for their ability to eke out some reverse swing when New Zealand batted a second time.
"When you know you've got favourable conditions it's not always easy to execute as well as they did for as long as they did," McCullum said of his side being reduced to 51-5 inside the first 12 overs in the first innings.
"(Then) the way they went about the second innings on that wicket, they identified pretty quickly the ball wasn't swinging conventionally.
"Once the ball is reverse swinging, its another thing to apply pressure and execute it and they did that outstandingly well."
Smith was keen for his side to build on their win, with pace bowler James Pattinson likely to come back into contention, while seamer Peter Siddle would need tests on a back spasm.
"We have played some very good test cricket over the last couple of months," Smith said, referring to series wins over New Zealand and West Indies.
"Obviously to win both series at home and start this one the way we have has been really satisfying and there is a lot of improvement left in us, but I think we're going in the right direction."
Australia top the one-day international rankings, are second on the test list but a lowly eighth on the T20 table ahead of next month's ICC World Twenty20 in India.
(Editing by John O'Brien)
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