Lote Tuqiri gave Australia an early lead but then the All Blacks pack showed their power with two tries from hooker Keven Mealamu helping them to a half-time lead of 14-7.
Scott Fava cut the deficit soon after Richie McCaw had looked to have given the All Blacks a winning advantage, but in truth the Wallabies' comeback hopes were long gone by the time Isaia Toeava sealed a comfortable win late on.
Rocky Elsom was yellow-carded for repeatedly failing to roll away from the tackle late in the first half, and the All Blacks took full advantage.
Connolly felt the Wallabies number eight's sin-binning had a big bearing on the outcome.
"It was significant," said Connolly. "When you play a man short it has been shown you end up coughing up points and we gave away 14 points.
"I thought he was a bit unlucky to go. I couldn't quite see the replay on the screen but I thought we were a bit unlucky. He [Elsom] made the tackle and slipped past and he was just caught on the wrong side.
"They were very physical but we knew what to expect," he added. "We knew they would be a threat to us.
"But all of their four tries we contributed to and you just can't afford to slip up against an opposition of that calibre.
"New Zealand are a great side and they gave us very little. We are very disappointed. There was no lack of effort but good sides will nail you to the wall and they did.
"We went alright in the first couple of scrums then fell away in the rest of the first half."
The All Blacks took their time to gel but once they clicked they were far too strong for the Wallabies, and coach Graham Henry believes their victory could have been more emphatic.
"I think it took us a while to settle," he said. "I think the first 20 minutes the guys were finding combinations and getting used to playing together again.
"I think we had the better of the last 60 [minutes] and it could have been a higher score."
Australia will be looking for an improved performance when they take on South Africa in Brisbane next weekend in the second match of the series.