Boult and Southee bowl brilliantly against England to ensure a cracking contest continues today.
England rattled New Zealand's cage late in the day but New Zealand are in charge of the deciding test at Eden Park with two days remaining.
A 239-run first innings lead will do that. New Zealand start the penultimate day of the ANZ international series today on 35 for three, ahead by 274 overall and on a pitch which offered more yesterday to the bowlers than the previous three days had.
They took control through their seamers Trent Boult and Tim Southee, who were outstanding, finding movement where England hadn't.
Left armer Boult finished with career-best figures of six for 68 off 25 overs as England fell for 204. Southee, equally impressive, took three for 44.
They had the tourists 92 for five at lunch, and although Matt Prior, through aggressive means, and the more sedate Joe Root dominated the middle session, Boult roared back after tea with three wickets in 13 balls.
Three lbw decisions in the morning session - two won with New Zealand referrals, the other, Ian Bell, would have survived had he chosen to seek a second opinion - undermined England's innings and may yet prove to be the decisive period in the match.
Both Boult and Southee showed their capabilities yesterday.
"The last couple of days it felt pretty good. There's a little bit of swing out there, which is nice," Boult said. "There are a couple of signs of a little bit of up and down going on [in the pitch]. That's always pretty exciting to see from a bowling point of view."
Operating to a fuller length than England had certainly helped.
"That's our traditional length, myself and Tim. There's no doubt you've got to give it every opportunity to swing, and by pitching it up you're encouraging that."
Prior quipped that maybe "they had a better box of balls". However, the wicketkeeper insisted England would fight on.
"We know we have the skill and ability to get something out of this game. There's still a lot of cricket to play," he said.
Prior has had a strong series and his 73 yesterday revived a sagging innings. From his dismissal, clipping Neil Wagner to point to his mortification, England lost five for 31 - the last four for four after tea.
Unlike England at Wellington a week ago, New Zealand didn't enforce the follow on. They want to bowl last. In any case their seamers needed to put their feet up.
Broad and James Anderson then put the skids under New Zealand, trimming their second innings to eight for three, before Peter Fulton and Dean Brownlie negotiated the final hour together.
This has been a cracking contest, most of which New Zealand has bossed. They can shut the door on England today.
New Zealand haven't won a home test against a major nation since November 2009 - Pakistan at Dunedin. The past 11, excluding Bangladesh and Zimbabwe, have produced six draws and five wins. England have scored over 300 to win a test just three times, but this is no time for New Zealand to get ahead of themselves. However, smart, assertive cricket over the next two days can bring a rich reward.
Best test figures for New Zealand v England
* 7-74: Lance Cairns, Leeds, 1983
* 7-143: Lance Cairns, Wellington, 1984
* 6-26: Richard Hadlee, Wellington, 1978
* 6-52: Chris Cairns, Auckland, 1991-92
* 6-53: Hadlee, The Oval 1983
* 6-54: Daryl Tuffey, Auckland, 2001-02
* 6-67: Jack Cowie, Old Trafford, 1937
* 6-68: Trent Boult, Auckland, 2013.