The visitors battled to reach 5-225 by the close at Headingley - but only after Henry Nicholls had been barely able to believe his misfortune when he got out in the most freakish manner just before tea on Thursday.
The Black Caps' batter's straight drive careered into the outstretched bat of non-striker Mitchell, who was attempting to take evasive action at the other end, and rebounded all the way across to mid-off where Alex Lees was alert enough to take a simple catch.
There were looks of bewilderment across the faces of England's players - notably, bowler Jack Leach, who claimed his second wicket of the day - as Nicholls trudged off for 19, leaving New Zealand stuttering on 5-123 at tea.
That, however, brought together Blundell and Mitchell, who have shared partnerships of 195 and 236 already in this three-match series in which England hold an unassailable 2-0 lead.
They put on an unbroken sixth-wicket stand of 102 to be there at stumps, giving the tourists renewed hope of a consolation victory.
Blundell was on 45 while Mitchell was on 78 and closing in on his third century of the series.
Mitchell capitalised on being given a life when on eight by England captain Ben Stokes, who chose to not review when the New Zealand batter was trapped shin height in front of the stumps from a full delivery by Matthew Potts that swung in.
Had Stokes reviewed, Mitchell would have been on his way and the tourists would have been 5-97 and without their best player over the past month.
Stuart Broad set the tone for an impressive bowling performance by England, ending the day with figures of 2-45 off his 17 overs as the pacer relished the responsibility of being the senior man in England's attack in the absence of fellow veteran Jimmy Anderson.
After New Zealand won the toss, Broad removed Tom Latham for a duck off the sixth ball of the day and got another edge to remove Kane Williamson for 31 before lunch.
Between those wickets, the first ball of Leach's left-arm spin drifted and straightened up, trapping Will Young lbw for 20.
New Zealand had plenty of work to do at 3-65 at lunch, but Devon Conway (26) had only got started when he dragged a seaming delivery from debutant Jamie Overton onto his stumps with a loose drive.
Overton was applauded from the balcony by twin brother Craig, who was edged out of the starting team but came on briefly as a substitute fielder early in the day to ensure the siblings were out there at the same time for an over or two.
"We used to talk about this as kids — playing Test match cricket is the ultimate goal," Jamie Overton said. "He was delighted for me."
Mitchell played with the same energy he has shown since touching down in England, sweeping and pulling well before the second of two spiralling sixes off Leach took him past 50.
Meanwhile, Stokes stayed mysteriously out of the attack, perhaps to save energy with the series won and with a one-off Test against India coming up next week.
