A fifth Test ton for BJ Watling (100no) ??? the first by a New Zealander in Leeds ??? helped the tourists close on 338-6 after a dashing Stuart Broad knock (46 from 39 balls) saw England match the tourists??? first-innings score of 350.
The Black Caps’ healthy advantage, swelled by fifties for Martin Guptill (70) and Brendon McCullum (55), means the hosts will have to secure at least the second-highest successful run chase in a Headingley Test to wrap up the series.
Play began 15 minutes late on Sunday morning due to rain and England were caught napping as Ian Bell (12), Jos Buttler (10) and Moeen Ali (1) all edged Tim Southee (4-83) into the slip cordon early on.
Alastair Cook???s charges had sunk from their overnight 253-5 to 267-8 at that point but brisk hitting from Broad ??? who belied his recent batting woes with five fours and a six ??? and Mark Wood (19) added a quick-fire 51 runs.
Wood clubbed three boundaries of his own before feathering behind, while James Anderson (10no) ??? now just six runs shy of 1,000 in Tests ??? put on 32 with his long-time bowling partner.
Broad was the final man out, cleaned up by Matt Henry four runs short of an 11th Test fifty, as two sides recorded the same first-innings score in a Test for just the eighth time in history.
Paceman Broad (2-52) showed his ability with the ball, too, having Tom Latham (3) and Kane Williamson (6) snaffled behind by wicketkeeper Buttler shortly after lunch as the Black Caps slipped to 23-2.
However, Guptill and Ross Taylor (48) then scored at a typically speedy rate during a 99-run partnership, the former securing his 14th Test fifty ??? and second of the series ??? by smashing Moeen back over his head for six.
Taylor and Guptill both fell to Wood (3-64) ??before the tea interval ??? hitting to Ben Stokes at extra-cover and Joe Root at slip respectively ??? but McCullum and Watling flourished after the break.
McCullum, playing in unusually placid fashion for much of his knock, became just the second New Zealander, after Stephen Fleming, to pass 6,000 Test runs during his 121-run tandem with Watling.
The 33-year-old was banished lbw by the impressive Wood after compiling his 29th Test half-century and Anderson ended another enterprising dart from Luke Ronchi (31 from 23), before Watling reached three figures in the penultimate over of the day, off 136 deliveries.