New Zealand were 155-6 at the close, ahead by just four runs and with only four wickets remaining. Daryl Mitchell was not out 31 with Nathan Smith on one.
England made 499 in their first innings, an imposing 151-run lead, led by Harry Brook's 171.
"We're in an amazing position to go on and win the game," said Brook, who benefitted from five dropped catches. "I had a lot of luck, didn't I. and thankfully I cashed in."
New Zealand were hurting at the fielding errors with Brook adding 153 after he was first dropped on 18.
"Fielding's something we pride ourselves on and that's the part that hurts," said seamer Matt Henry.
"We work hard on it and it's usually where we turn a game on its head. So obviously it hurts when chances go down."
Brook's knock was supported by 80 from captain Ben Stokes and 77 from Ollie Pope.
It was then the turn of England's seamers as New Zealand's salvage attempt faltered at the start with Tom Latham caught by Brook for one to give Woakes his first wicket in the third over.
Devon Conway was on eight when his attempt to pull Carse to the boundary was miscued to Gus Atkinson at mid-on and New Zealand were tottering at 23-2.
Kane Williamson and Rachin Ravindra set about stitching the innings together as they put on 41 for the third wicket before Ravindra went for 24.
The left-hander could not resist pulling a short ball from Carse but failed to get on top of the bounce and was caught by Jacob Bethell waiting at mid-wicket.
Woakes returned to remove Williamson and Tom Blundell in successive balls to end the day with 3-39.
Williamson, so often called upon to rescue his country, was unruffled until on 61 -- his second half-century of the Test -- he missed a straight delivery and was trapped lbw.
With his next ball, Woakes found the faintest edge of Blundell's bat but he was denied a hat-trick by Glenn Phillips who pushed a short ball towards the covers.
Phillips progressed to 19 before he was given out lbw to a Carse delivery that nipped back.
New Zealand sought a review but replays showed the ball was skimming the bails and the decision was the umpires call and Carse had 3-22.
Masterful
England resumed the day at 319-5 and raced along at six an over before being all out after lunch.
A masterful Brook, who benefitted from five of the eight catches spilled by New Zealand, led England's rescue after they were left reeling at 71-4.
The only good news for New Zealand was that while they dropped two more catches on day three, they successfully held five.
Brook relished his lives. He smiled and looked to the heavens when Phillips spilled a chance on 147, having been dropped the previous day on 18, 41, 70 and 106.
Four overs after his last reprieve, he brought up his 150 in imperious fashion, charging down the wicket at Tim Southee and driving him to the cover boundary.
But when New Zealand found a way to make catches stick, Brook's innings, which included 15 fours and three sixes, came to an end.
A good length ball from Matt Henry found a faint outside edge and wicketkeeper Blundell grasped the opportunity.
The applause as Brook left the ground was as much for New Zealand holding a catch as it was for a magnificent innings.
Woakes (one) was out on the second ball he faced with Latham scooping up a low-level chance at second slip.
Smith ended Atkinson's breezy innings with Phillips taking a smart catch at long leg.
Henry was the most successful New Zealand bowler with 4-84. Smith took 3-141.
