New Zealand was 168-5 for an overall lead of 137 runs when rain brought an early closure with 17.1 overs remaining in the day.
Jamie Overton missed out on a dream hundred in his debut test but Jonny Bairstow's masterful 162 gave England a slender first-innings lead of 31 runs when it was all out for 360 just before lunch.
Tom Latham (76) and captain Kane Williamson (48) raised New Zealand hopes of setting a big target in the last test of a compelling series until England hit back with four wickets in the last session -- claiming the last three for only nine runs.
That left New Zealand's best batters in the series, Tom Blundell on 5 and Daryl Mitchell on 4, at the crease.
Latham and Williamson batted solidly against pace and dominated the spin of Jack Leach before Stuart Broad induced a healthy outside edge of Latham's bat which Joe Root couldn't hold at first slip late in the second session.
There was virtually no lateral movement for the fast bowlers and Latham was quick to punish anything pitched in the slot. Both batters also played some impressive sweep shots and combined in hitting 18 fours.
Latham and Williamson nearly played out the entire middle session and took their team to a comfortable 125-1 at tea. Then fast bowler Jamie Overton struck off the first ball after the tea break.
Latham, who hit 12 boundaries, was tempted outside off stump and edged behind.
Devon Conway (11) didn't look comfortable against pace and England struck immediately after a short rain break when Ollie Pope plucked a brilliant one-handed low catch at forward short leg off Joe Root's offspin.
Matthew Potts, who claimed Will Young for 8 just after lunch, got the key wicket of Williamson, who edged behind after nearly three hours.
Leach snapped up a regulation return catch from Henry Nicholls and New Zealand slipped to 161-5.
With still two days left in a series already lost to New Zealand, it will be hoping Mitchell and Blundell can post another century stand to set a competitive chase for England.
Earlier, England's lower order continued to prosper after it resumed from 264-6 overnight.
Jonny Bairstow made 162 and Overton 97, and their record seventh-wicket stand of 241 lifted England from a precarious 55-6 to 360 all out before the hosts were bowled out before lunch.
Overton was overcautious against the pace after resuming on 89, and drove Trent Boult for his first boundary of the day and 13th overall.
Bairstow, 130 overnight, raised England's second fastest 150 in a test off 144 balls when he drove Neil Wagner through the covers for two runs.
Boult, who sliced through England's top order on Friday, finally broke the stand when Overton edged a low catch to the slips after the left-arm fast bowler changed the line by coming round the wicket. Overton's brilliant counterattack came off 136 balls and featured 13 fours and two sixes.
Boult finished with 4-104.
Broad played a superb cameo of 42 off 36 balls that included sixes against Boult and Wagner. Tim Southee (3-100) clean bowled the left-hander.
Bairstow's second successive blistering knock of the series finally ended when he came down the wicket against the spin of Michael Bracewell and Boult took a smart catch at long off. Bairstow received a thunderous applause from England supporters on his home ground.
Bairstow's blistering knock off 157 balls featured 24 fours, and rescued England hopes of sweeping the series against the world test champion.