Bates and Dodd spent the ensuing five overs fending at a torrid South African assault, the New Zealand captain having found just a single before Kapp rocked her side further.
Bowling Bates at the close of the eighth over, a wicket maiden to Kapp had her holding supreme early figures of 4-4-0-1. A tiring Khaka completed her first spell for 6-3-16-2 after Dodd finally picked her off for a couple of boundaries, but the ascendancy was all one way.
"It was hard work at the beginning, having lost a few wickets at the other end," Dodd would mention later. "But it was a good pitch, and I kept thinking to myself that if I got myself in, the runs would come. The biggest focus for me was keeping a clear head to stick with my game plan and my routines - and today it paid off."
Dodd and leading series runmaker Amy Satterthwaite (again batting down at five) stitched together a recovery, their 84-run stand for the fourth wicket the highest of the White Ferns' innings.
Calm Satterthwaite had reached 42 from 47 balls when disaster tolled again, however, run out in the 27th over after Dodd smashed a straight drive only for bowler Suné Luus to tip it on the non-striker's castle with Satterthwaite out of her ground. At 94 for four, that meant new batsman Katie Perkins had an imminent mission nursing Dodd (38* at the time) to a maiden half century - not to mention getting the overdue New Zealand hundred up on the tins.
Dodd's previous best knock in 10 matches had been just 22, but 106 balls of concentration delivered that breakthrough fifty at a moment that could not have been better chosen for her side.
She would be stumped on 52 in the 32nd over off Dané van Niekerk; meanwhile Perkins knuckled down for her eventual 34 off 45 balls, going on to share a 43-run sixth-wicket stand with Sam Curtis that helped the White Ferns salvage a respectable 208 for eight from their full 50 over allotment.
Senior strike bowler Lea Tahuhu had come back into the XI for Holly Huddleston, while Curtis had returned for an ill Thamsyn Newton. Using her pace, Tahuhu got the visitors off to a strong reply, quickly zeroing in on South African openers Lizelle Lee and Andrie Steyn, both caught behind as keeper Rachel Priest set to work on four dismissals in an innings.
After scoreless Dinesha Devnarain had also fallen, off leg-spinner Erin Bermingham, at 28 for three South Africa found themselves back in the familiar territory of needing their star players to bail them out. But they were further hobbled when a concussed Marizanne Kapp was forced to retire hurt before she had even got off the mark, having been struck on the helmet by Tahuhu.
With Lara Goodall (dislocated finger in the field) already having been invalided out of the order, the cheap dismissal of influential van Niekerk was a major second breakthrough from Bermingham, the home team's task now falling on the shoulders of first drop Mignon du Preez and Luus, who would prove the only serious contributors to the South African cause as the only batsmen to reach double figures.
Luus offered a gutsy 27 off 52 before Bates (3-17 off five overs) struck with consecutive balls in the 26th over; Luus caught by Katie Perkins and Khaka given some payback with a golden duck.
Experienced du Preez battled on for her ninth ODI half-century against tight bowling from the White Ferns attack, but the required run rate was escalating and her dismissal, for a 90-ball 62, gave Bermingham three for 28 as the win was confirmed with more than 14 overs to spare.
Overall, it was a remarkable fightback from the White Ferns who kept their composure as they dug themselves out of a predicament.
The sixth ODI in the seven-match series will also now take place at Paarl's Boland Park, having been moved from Coetzeburg Oval.
Play begins at 9pm NZT this Saturday, 22 October, the White Ferns leading 4-1.
