But it was far from plain sailing for Graham Henry's men, who mounted an uncharacteristic rearguard action to preserve their slender lead.
After a shaky start, the Australia pack gained the upper hand in the second half but their backs were consistently thwarted by some last-gasp New Zealand defending.
Connolly said: "It was good, I mean we had a couple of very rocky [scrums] at the start, but overall I think we stood up pretty well and I think those guys will definitely get a lot of confidence out of that.
"It was one of our better forward performances. I'm very proud of the effort they put in, and we've got to keep working on the reasons why we lost the game. We turned the ball over in attacking areas."
It was his side's failure to take their chances that most upset Connolly, who added: "It was definitely a game we thought we could have won.
"New Zealand had a couple of chances to put us away in the first half and didn't, and we had two or three in the second half and didn't.
"We're gutted that we lost the game but I think we showed we're not that far away."
George Gregan, who equalled John Eales' record of 55 Tests as captain of the Wallabies, agreed with Connolly.
"We're bitterly disappointed, I think everyone in the room's are aware we had our chances to win it," he said.
Gregan had spoken before the game about the need for Australia to grasp any opportunity the in-form All Blacks presented in the match, but he rued their inability to convert those chances into points.
He added: "The second half probably provided about three or four opportunities to put them away which we didn't, just a little bit of execution let us down.
"When you're coming up against a strong defensive unit like that, you've got to take those opportunities."
Australia have a week to bounce back before facing South Africa on Saturday at Sydney's Telstra Stadium.