New Zealand's hopes of survival took a heavy blow just before tea when key batsman Kane Williamson was out leg before wicket for 59 in 73 balls.
He tried to pull offspinner Nathan Lyon and was given out by umpire Richard Illingworth.
The ball struck first innings centurymaker Williamson just above his waistband but replays suggested the ball would have clipped the top of the bails, thus backing Illingworth's decision.
It was the third time Williamson had made a century and 50 in the same test, joining John R. Reid and Nathan Astle among New Zealand players to that achievement.
He went to his 50 with a fine stroke to the mid wicket boundary off Mitchell Johnson, shortly before his dismissal.
The other batsman out in the session was opener Martin Guptill.
He had knuckled down impressively, albeit surviving two dropped catches on 0 and 7, before being finally winkled out by offspinner Nathan Lyon for 23, on the back foot and edging a catch to Steven Smith at slip.
The usually assertive Guptill had batted 186 minutes, and faced 133 balls in a doughty performance.
By contrast, Taylor, who arrived at the crease on a pair, had an edgy start, before plundering a couple of boundaries through square leg and mid wicket.
New Zealand's highest successful fourth innings run chase was 21 years ago when they scored 324 for five to beat Pakistan in Christchurch.
Australia were the losers in the two highest winning chases in test history - the West Indies' 418 for seven at Antigua in 2003, and South Africa's 414 for four at Perth in 2008.
By David Leggat in Brisbane
