Nine wickets down, England's lead at that stage stood at 68, but some unorthodox hitting from final-wicket pair Neil Wagner (24no) and Boult (16) brought the Black Caps closer to parity before Broad finally wrapped up the innings.
England's experienced opening pair of Stuart Broad and James Anderson ran through New Zealand's top order after they posted 307 in their first innings.
England made the new ball count with Broad (4-38) having Tom Latham (0) caught behind in the second over and in the fifth over James Anderson (2-43) removed Jeet Raval (5) in the same fashion.
Captain Joe Root is at the crease on 30 and probably facing a key declaration decision tomorrow.
Alastair Cook failed again but England remained in a strong position in the second cricket test against New Zealand on Sunday, leading by 201 runs with with seven second innings wickets remaining after tea on the third day.
It left New Zealand 239-9 and England poised to clean up the tail but Boult (16) and Wagner (24 not out) had other ideas as they cut the deficit to 29 runs.
Stuart Broad and Jonny Bairstow put England in charge of the second Test in Christchurch as the New Zealand disintegrated to be 32 for four at lunch on day two in reply to the tourists' first innings 307.
Late Homer by Blue Jays' Solarte Sends Yankees to First Loss
Ellsbury could be available by next Thursday, at the earliest, when the New York Yankees host the Baltimore Orioles . Ryan Tepera (1-0) pitched one inning for the win and Roberto Osuna finished for his first save.
Dawid Malan departed for a golden duck in Boult's following over, trapped plumb LBW, before an awkward 111-ball vigil from opener Mark Stoneman (35) ended when he snicked Southee to second slip.
Alastair Cook still has the appetite for Test cricket despite a hard tour to New Zealand, according to England batting coach Graham Thorpe.
Bad light stopped play with New Zealand, who lead the two-match series 1-0, closing on 192-6, trailing by 115 runs. Resuming at 97 not out, he took three singles to reach his hundred from 164 balls with 11 fours and a six before being the last man out.
'There are areas of the Test side where you're not going to say everyone is guaranteed their places, we're constantly trying to get more out of players who are in the XI, ' added Thorpe.
It was Colin de Grandhomme and BJ Watling that ensured the home side did not suffer the same fate as England did in the first Test. The hosts won the first test at Eden Park in Auckland by an innings and 49 runs.
'They are still developing within their Test careers, but I was pleased with the way they went out and got themselves into good positions, and didn't back off'.
"It's a massive session [today]".