The win seals a series victory for the home side with just one game to play.
The Black Caps skittled the tourists for 175 in just over 35 overs and, despite some fine pace bowling from Jofra Archer, Mitchell (56) again steered the hosts across the line as he had in their victory in the series opener in Tauranga on Sunday.
Seamer Tickner was brought into the squad on Sunday as a replacement for the injured Kyle Jamieson and got his first international start since May 2023 when Matt Henry was ruled out of the match just before play.
The 32-year-old wasted no time in making his presence felt at Seddon Park by strangling Joe Root down the leg side to oust the former England skipper for 25 before ripping through the tail to claim 4-34, his best figures in the ODI format.
Mitchell Santner's spin accounted for dangerman Harry Brook after a run-a-ball 34, while Jamie Overton top-scored for the tourists with 42 from 28 balls before he holed out.
Archer (3-23) trapped Will Young in front for a duck on the fourth ball of New Zealand's innings and the England quick's pace made the early part of the chase distinctly uncomfortable for the home batters.
Kane Williamson eked out 21 from 39 balls before dragging on an Overton delivery to bring Mitchell out to join Rachin Ravindra with 134 runs still required for victory.
Opener Ravindra picked out a fielder at deep fine leg with a pull shot early in Archer's second spell to depart for 54 and Tom Latham quickly followed to give Adil Rashid his first wicket.
Archer removed Michael Bracewell for five with his final delivery of the match but skipper Santner made 34 not out and Mitchell secured the winning runs in the 34th over with a thick outside edge for his sixth boundary.
