Broad's 46, his highest Test score since he made 47 against India at Trent Bridge last year, helped England make 350 -- the exact same total as New Zealand posted in their first innings.
In the four overs possible before lunch, New Zealand took their second innings score to seven without loss, a lead of seven runs, in a match they had to win to level the two-Test series at 1-1 after their 124-run defeat at Lord's.
Tom Latham was three not out and Martin Guptill four not out.
But Broad, dropped on nine, and Mark Wood (19) repaired the damage during a fluent ninth-wicket stand of 51 in just 49 balls.
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Broad and James Anderson (10) put on 32, with England's last two wickets adding 83 runs in all.
Southee led New Zealand's attack with four for 83 in 30 overs.
England had been cruising while left-handed openers Adam Lyth (107) and England captain Alastair Cook (75) were sharing a first-wicket stand of 177.
As a result, Ian Bell (12 not out) and Jos Buttler (six not out) faced the tricky task of confronting New Zealand's attack when play started Sunday.
But the struggling Bell had failed to add to his overnight score when, to Southee's second ball of the day, he was drawn into driving loosely at an outswinger and edged low to Mark Craig at second slip.
Buttler (10) and Moeen Ali (one) then both fell tamely as they nicked Southee to the slip cordon.
But off-spinner Craig broke the partnership when, with his sixth ball, he had Wood caught edging an intended cut to wicket-keeper Luke Ronchi.
Anderson, however, drove Craig through extra-cover for a textbook boundary and next ball swept him for another four before Broad hooked fast bowler Matt Henry for six.