At stumps at Eden Park in the series-deciding final Test, New Zealand were three for 35, leading by 274 after dismissing England for 204.
Opener Peter Fulton, who scored his maiden Test century in the first innings, was unbeaten on 14 which included three boundaries, with Dean Brownlie on 13.
In a chaotic final session of play, eight wickets fell for 39 runs as the England tail collapsed and the New Zealand top order followed suit, increasing the chances of a decision in the deadlocked series.
The rain-abbreviated first two Tests were drawn but only fine weather is forecast for Auckland over the next two days.
New Zealand decided not to enforce the follow on after ripping out the England tail after tea as they gambled on bowling last on a deteriorating wicket.
But the strategy appeared to backfire as Hamish Rutherford fell on the third ball he faced to start a collapse that reduced New Zealand to eight for three after eight overs before Fulton and Brownlie steered them through to stumps.
The New Zealand batting failure came after their strike bowlers Tim Southee and Trent Boult had ripped through the England tail, taking the last five wickets for 31 runs.
The eight dismissals after tea occurred in the space of 18 overs as the wicket which supported the New Zealand batsmen to a first innings 443 began to show assistance for the bowlers.
For New Zealand, left-armer Boult produced his best Test figures of six for 68 while Southee finished with three for 44.
Broad had the remarkable figures of two for seven from seven overs for England.
He removed Rutherford for a duck with a rising delivery that was flicked to Ian Bell in the gully and trapped Ross Taylor (three) leg before wicket.
Kane Williamson had scored one run from 11 balls when he chopped a wayward delivery from James Anderson on to the stumps
England, who started the day at 50 for two, staged two spectacular collapses in an innings that was boosted by the single partnership of Matt Prior and Joe Root.
They lost three early wickets for 22 runs as Nick Compton (13), Bell (17) and Jonny Bairstow (3) struggled to cope with the morning swing generated by Southee and Boult and all fell leg before wicket.
At 72 for five and their backs firmly against the wall, Prior and Root set about salvaging the England innings and produced a 101-run partnership for the sixth wicket.
Prior was dropped by Brownlie at second slip on 24 and went on to top score for England with 73 when he was caught at point slashing at a wide Neil Wagner delivery just before tea.
Root produced England’s second highest score, working his way to 45 when he was bowled by Southee to end the innings.
Broad caused a brief flurry of activity on the scoreboard, and distorted Boult’s figures, when he belted 16 runs in the space of four deliveries.
But the bowler had the final word when Broad’s next attempt to reach the boundary saw him caught at cover by Rutherford.