Comfortable at 3-202 in their second innings, England lead by 231 and will target a declaration at some stage on Monday, assuming they see off the second new ball's morning arrival without drama.
England have had a long and tough winter schedule but they enjoyed a good day in Christchurch with both bat and ball.
But things are looking wholly more positive now, with Joe Root and Dawid Malan at the crease looking to kick on when play resumes.
He was the last man out, giving Trent Boult his fourth wicket of the innings.
England's all-time record Test runscorer has mustered just 23 in his four innings in this series - a personal low. Stoneman was dropped twice on 48 and 57 and reprieved on review after being given out caught behind, while Vince was lucky to not be caught twice off leg-spinner Ish Sodhi.
It also ensured opener Stoneman and number three Vince should retain their positions for the start of the summer to come.
However, they fell in similar - and frustrating - fashion.
But apart from Wood and Bairstow there was little to enthuse about as Boult and Southee swung and seamed the ball with precision.
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Petersburg Fire Rescue was called to Tropicana Field to put out what team officials called "a small grease fire" on the 300 level. Sale was 4-4 against the Rays entering the 2017 season but dominated them in helping Boston win an 11-8 season series.
Broad proved the value of his hours alone in the Trent Bridge indoor nets between tours when he took four for 38 against New Zealand on day two of the second Test, moving past the great Curtly Ambrose to 14th on the all-time global list with 406 wickets. While Watling's contribution on Sunday was brief before he was bowled by Anderson, his overnight partner Tim Southee went on to make 50, his fourth test half century.
Broad picked six wickets for 54 runs and was well supported by Anderson (4/76).
Bad light stopped play with New Zealand, who lead the two-match series 1-0, closing on 192-6, trailing by 115 runs.
"It's turned into a very nice batting track, there's a lot of wickets for us still to take".
"I had to improve - and over that short period of time, I think I have".
'I'm sure both of them would have loved to have got hundreds, and gone on for their own confidence. "He looked good in the first two or three Tests of the Ashes without scoring many runs".
In fact, England haven't won an away test match since beating Bangladesh in the first test by just 22 runs in Bangladesh back in October 2016.
"Watling didn't really give us a chance".