Three first-half penalties from Aaron Cruden left the All Blacks with a 9-3 lead at the end of a first half which saw limited opportunities for both sides.
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However, Wyatt Crockett was sent to the sin bin shortly before the break, leaving New Zealand a man short at the start of the second half.
Australia made the most of their advantage, setting up Kurtley Beale for two penalties to level the scores in the 55th minute.
Cruden restored the world champions' lead just four minutes later, but discipline cost the team again when Beauden Barrett was shown a yellow card.
Australia had a penalty as a result of Barrett's error, which allowed Beale to split the uprights again, getting the Wallabies back on level terms to secure the draw.