England 303 and 122 for 9 (Wood 29, Wagner 3-18, Henry 3-36) lead New Zealand 388 (Young 82, Conway 80, Taylor 80, Broad 4-48) by 37 runs
England were left reeling at 3 for 30 as Henry tore through their top order either side of tea and by the close, they were nine down and just 37 runs ahead. Their fragile middle order was exposed once more and New Zealand were on the cusp of victory - although not enough to take the extra half-hour at the end of the day to try and finish it off.
Having resumed on 229 for 3, still 74 runs behind, New Zealand lost their remaining seven wickets for 96 runs, but that wasn't quickly enough for England, who collapsed to 76 for 7 still nine runs adrift, only clawing their way ahead via an eighth-wicket partnership between Mark Wood and Olly Stone.
Bearing in mind that New Zealand were resting spearheads Tim Southee and Kyle Jamieson ahead of next week's WTC final, a theme that had developed surrounding the tourists' incredible depth became outright dominance as England had no answers.
Ross Taylor batted with greater fluency than on the previous day, pushing his overnight score on from 46 via a series of boundaries, including a deft sweep off Dan Lawrence to bring up his fifty. He then hit back-to-back fours off James Anderson through the point region, with the second looking more intentional than the first.
Taylor was dropped on 68 hooking to long leg, where Sam Billings grassed the chance but he was eventually out nicking Stone through to wicketkeeper James Bracey for 80, the third New Zealand batter and fifth overall this match to be dismissed in the 80s.
Bracey, the England wicketkeeper playing his second Test, had a moment to forget a short time later when he dropped Tom Blundell off Stone with the batter yet to score.
Wood was bowling with sharp pace for reward as Blundell and Henry Nicholls withstood the pressure. But that was only until Nicholls was struck on the helmet attempting to pull a rapid bumper. No sooner had he been given the all-clear to continue, Nicholls gloved Wood's next delivery down the leg side and Bracey held on.
Anderson had toiled for 24.2 overs before taking his first wicket of the match, bowling Wagner for a duck and, when Wood had Henry out lbw, New Zealand's lead was just 50 with only two wickets in hand.
Stuart Broad mopped those up, including the wicket of Blundell, caught by Joe Root, who let out an almighty shout directly at the ball after he held on at slip, having dropped one the previous day off Stone that would have dismissed top-scorer Will Young on seven. Broad remained the pick of England's bowlers with 4 for 48 from 23.1 overs.
But the lead was 85 by that point and, when Henry had Rory Burns out for a second-ball duck, attempting to drive and edging to Tom Latham at second slip, the danger signs were there. Burns had been England's in-form batter this series and it would fall to someone else this time. As it turned out, not even Root could come to the rescue.
Henry had Dom Sibley caught by Daryl Mitchell at third slip shortly before tea and, after the break, Zak Crawley's lean run continued when he fell lbw to Henry for 17 Ollie Pope rattled along to 23 off just 20 balls but he was struck on the knee roll by a Wagner inswinger and sent on his way with England still 27 behind.
Lawrence, who has impressed among the relative newcomers to this England side with a fifty on debut in Sri Lanka in January, scores of 46 and 50 against India in Ahmedabad and an unbeaten 81 in the first innings of this Test, was Wagner's second scalp, caught behind without scoring.
Bracey, out for a duck on debut at Lord's and again in the first innings here, managed a wry smile as the Edgbaston crowd roared when he got off the mark and they were equally enthusiastic when he pulled Trent Boult through midwicket for four.
His relief was brief, though, when Patel was re-introduced into the attack and struck with his third ball when Bracey moved across his stumps and ended up pressing the ball into middle with his glove.
Where there was Root there was hope, even if he had faced 54 balls to reach double figures. But his attempted cut off Patel produced only a top-edge through to Blundell.
What remained for home fans was another entertaining knock from Wood, who had put on a show on the second morning to reach 41 with some big hitting. This evening he took England into a five-run lead with a slog-sweep off Patel for six over deep midwicket. Two fours off as many balls in Patel's next over brought up England's hundred but they were seven wickets down and only 15 ahead.
Wood's partnership with Stone was worth 44 before Wood skied a Wagner short ball almost directly above his own head and Blundell had aeons to set himself beneath it, gloves at the ready.
Boult rearranged Broad's stumps in the last over of the evening, but Anderson survived the remaining four balls to ensure the match would see a fourth day.