Broad ensured first innings parity for England on the third day and then reduced New Zealand to 23 for two in their second innings.
But in what is now typical New Zealand style, Martin Guptill (70) and Ross Taylor (48) added 99 in 87 balls for the third wicket.
However, both well-set batsmen fell in quick succession to Durham paceman Wood.
At tea, New Zealand were 143 for four, a lead of 143, in a match they had to win to level the two-Test series at 1-1 after their 124-run defeat at Lord's.
Broad's 46, his highest Test score since he made 47 against India at Trent Bridge last year, helped England make 350 -- the same total as New Zealand posted in their first innings.
He then took two wickets for four runs in 12 balls, with Tom Latham (three) and Kane Williamson (six) both caught behind.
Taylor should have been out for six when he edged an intended drive off James Anderson.
But, not for the first time this series, England dropped a slip catch, with third slip Gary Ballance failing to hold the chance above his head.
Meanwhile opener Guptill, one of the stars of New Zealand's run to the World Cup final, drove off-spinner Moeen Ali for six to complete a fine fifty.
But the pendulum swung back in England's favour when Taylor chipped the first ball of Wood's second spell to Stokes at extra-cover for an all-Durham dismissal.
Wood was again rewarded for pitching the ball up when Guptill's intended drive flew quickly to Joe Root, who held a sharp catch at third slip to end a 72-ball innings including seven fours and a six.
But Broad led a tail-end revival that saw England's last two wickets add a valuable 83 runs.
The struggling Ian Bell had failed to add to his overnight 12 not out when he was drawn into driving loosely at a Southee outswinger and edged low to Mark Craig at second slip.
Things might have been different for England had Broad, then on nine, not been dropped in the deep by Watling.
Instead Broad, who has a Test hundred to his credit but whose batting has suffered since he was hit on the head by India's Varun Aaron last year, continued to attack.
Broad hooked Matt Henry for six before he was bowled by the fast bowler to end a 39-ball knock that also included five fours.
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