Meanwhile, the West Indies closed the second day's play at 215/8, trailing the hosts by 158 runs in the first innings.
With West Indies 80-4 after Ambris left the crease, Shane Dowrich (0) followed his team-mate off the ground two balls later when he played a Wagner delivery to Henry Nicholls at short-leg.
New Zealand won the first match of the two-Test series by an innings and 67 runs in Wellington.
But Roston Chase made a half century in a 78-run partnership with Ramon Reifer (29) which delayed New Zealand's push for victory by nearly two hours.
After Ross Taylor slammed his 17th century to help New Zealand set a target of 444 runs, West Indies lost two wickets before stumps on day three and were restricted to 30 for two in the second Test in Hamilton.
The final session saw the former New Zealand captain upping the ante as he scored at run a ball to complete his 17th Test century and despite double strikes from Roston Chase (2-51) and Gabriel (2-51), New Zealand went on to reach 291 for eight when the declaration came.
And their predicament got worse early on the fourth day (last night Caribbean time) when captain Kraigg Brathwaite was dismissed 20 in the fifth over of the morning.
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But Hazlewood enticed a bottom edge and wicket-keeper Tim Paine did the rest to the unrestrained glee of the Australian team. Asked if he had high hopes at the start of day five, Root - who departed for 67 - told Channel 9: "Absolutely".
From that point it was the Wagner show - the star of the first Test producing a barrage of bouncers to leave the Windies reeling and in discomfort.
It seems like there is no end to West Indian batsman Sunil Ambris' tryst with hit wickets.
"I always thought I'd be just a white-ball player".
Taylor said he didn't get too nervous as he approached the three figure mark as the game situation was at the forefront of his mind.
The West Indies, who will have to beat their world record for the highest winning fourth innings score in test cricket - now 418 against Australia in 2003 - were 30-2 at stumps.