The hosts decided against forcing England to bat again after dismissing them for 204, then suffered their own wobble on the Eden Park pitch.
England face struggle to save Test despite New Zealand stumble
The potential series decider appeared destined for a draw, on a placid pitch, while New Zealand were piling up 443 on the first two days at Eden Park.
Yet England's inability to deal with conventional swing in their reply, then the Kiwis' decision not to enforce the follow-on and their own subsequent stumble to 35 for three left the game wide open once again.
Matt Prior (73) and Joe Root shared a hard-working century stand to try to rescue England from a perilous 72 for five.
Then once Prior fell just before the second new ball was taken, England lost their last four wickets for four runs to left-armer Boult (six for 68) and Tim Southee (three for 44).
New Zealand's batsmen caught the bad habit, and the hosts were a hapless eight for three at one stage to Stuart Broad and James Anderson. England got themselves into difficulties this morning when Southee won lbw verdicts against both overnight batsmen, Nick Compton and Ian Bell - and then Boult did likewise with Jonny Bairstow.
Compton's near strokeless innings last night had brought him 12 runs from 72 balls.
He mustered one more run, and lasted another five deliveries, today before Southee overturned umpire Paul Reiffel's initial not-out decision on DRS.
Southee had not struck for 89 overs in this series since he had Compton playing on to his stumps at the start of the first Test in Dunedin.
But he beat his forward-defence again here, and simulation demonstrated the ball would have clattered straight into leg and middle stumps.
Bell decided against a second review, after consultation with non-striker Root, when Southee got another one to nip in and beat the batsman stuck on the crease.
Bairstow, like Compton, was given not out - this time by Rod Tucker - pushing forward.
But Boult was already celebrating before even confirming his DRS intention, and it transpired Bairstow was plumb to another inswinger as England lost their third wicket for the addition of 11 runs.
Prior and Root had to dig in for an hour to lunch - a scenario which suited the latter's instincts better than his partner's.
Both fulfilled the brief, though, before Prior clicked into 'boundary' mode in the second session.
He had two moments of significant fortune, when a direct hit from substitute fielder Doug Bracewell at mid-on would have run him out on nine and then dropped low down at second slip by Dean Brownlie aiming a drive at Southee 15 runs later.
Root remained content to block and tire the Kiwis all the way to and beyond the second new ball - a moment Prior was not around to witness, having squirted a drive to point off Neil Wagner to end his 130-ball stay. Root was still not for moving, but received minimal support thereafter.
Broad clubbed Boult for two offside fours and a six pulled over midwicket in one over, only to spear the very next ball into the hands of cover.
Then Boult had Steven Finn edging to slip, and Anderson behind, leaving Root with only Monty Panesar for help.
The young Yorkshireman had little option but to try to hit England's way out of trouble, and was bowled aiming across the line at Southee - bringing an end to 176 balls and more than four hours of resistance, which had accrued just 45 runs.
England appeared to have put themselves in the tightest of spots, and highly unlikely to escape New Zealand with a series draw at best.
But their front-line bowlers had other ideas, and took the sudden rush of wickets to seven for 12 runs as the Kiwi top order fluffed their lines.
Only first-innings centurion Peter Fulton avoided the destruction as Broad had Hamish Rutherford splicing a simple catch to gully and Ross Taylor lbw playing across a straight one, and in between Kane Williamson edged Anderson down onto his stumps.
But in the 15 remaining overs - seven of the last eight maidens - neither team could make telling progress.
"Either way, you're behind the eight-ball - quite considerably," Broad admitted after the game.
"We know that, and we've got to front up to that.
"I think we showed we are doing that, with the 20 overs we've just bowled there. To have them three for 30 was a fantastic effort.
"The way our bowlers came out again, after obviously spending a lot of time in the field this Test match, they came in and fought hard."
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