Burns, who was dropped midway through the tour of India last winter following five single-figure scores in eight international innings, was given reprieves on 77 and 88 en route to a tenacious 132 out of England’s 275 all out.
The hosts had lurched from an overnight 111-2 to 140-6, with captain Joe Root falling to the first ball of the day before Southee used his guile and a hint of movement to snare Ollie Pope, Dan Lawrence and James Bracey.
But Burns showed admirable resolve amid the tumble of wickets at the other end, batting at a glacial pace for most of the day before accelerating after reaching three figures as he and James Anderson put on 52 for the last wicket.
Burns was last man out after nicking off to Southee, who finished with 6-43 in 25.1 overs, as the Black Caps claimed a 103-run first-innings lead, which they swelled to 165 after closing on 62-2 in their second dig.
Ollie Robinson, who has had a turbulent week on his England bow, bowled fellow debutant Devon Conway before trapping Kane Williamson in front and while a draw is now the favoured outcome, the hosts face the prospect of having to bat to save the Test on the final day.
A day-three washout was a hindrance for both sides but New Zealand were immediately in business on Saturday morning when Kyle Jamieson angled his opening delivery into Root, who could only edge low to Ross Taylor at first slip to fall for 42, ending a 93-run stand with Burns.
While Ollie Pope (22) showed some fluency after initially struggling as Jamieson probed away in the channel, the batsman overbalanced against Southee and was beaten all ends up, the not-out lbw verdict overturned on review.
Southee induced an error from Lawrence, whose loose waft at a wider delivery took the edge, before Bracey was castled between bat and pad in his first international innings, the pair both out for ducks as England lost three wickets on 140.
Southee’s morning spell of 9-4-18-3 had put New Zealand in control but Burns, despite adding just 13 runs to his overnight 59 in the first session, remained calm and stabilised England either side of lunch alongside Robinson.
There were few alarms until Burns was given a let-off on 77 in the over before the Kiwis took the second new ball, with BJ Watling missing a simple stumping after Mitchell Santner had beaten the advancing opener’s outside edge.
Robinson fell for a useful 42 after hooking to long leg as Southee sealed his place on the honours board, but the seamer gifted Burns another life when he spilled a chance at second slip after Neil Wagner had found a bit of purchase off the surface to take the outside edge.
Wagner castled Stuart Broad the ball after being deposited into the stands by the England veteran, leaving the hosts nine down and with Burns still nine adrift of a third Test hundred.
Anderson, though, was able to keep Jamieson at bay for a couple of deliveries before Burns squirted a couple through third man to bring up a 235-ball century, after which he cast off the shackles in a handy partnership.
Burns unfurled some glorious drives and even got on one knee to slog Wagner for six - his first maximum in Test cricket, to go with 16 fours in his innings - but he was unable to carry his bat as Southee wrapped up England’s innings before tea.
Anderson and Broad showed plenty of vigour at the start of New Zealand’s second innings and England’s record Test wicket-taker might have felt aggrieved not to get an lbw verdict against Tom Latham, the decision staying with the on-field decision because of ‘umpire’s call’ on impact.
England burned a couple of reviews on Conway, who made 200 in his first Test knock, but Robinson removed the umpire and technology from the equation when the New Zealand opener chopped on for 23.
Robinson, 27 - who issued an unreserved apology for historical tweets he posted in 2012 and 2013, when he was in his late teens, on the day he made his Test debut on Wednesday - then rapped Williamson on the pads. There was little enthusiasm when England sent the not out decision upstairs but technology backed Root’s query.
Williamson therefore trudged off for one off 19 balls, with Latham (30 not out) and nightwatchman Neil Wagner (2no) seeing New Zealand through to the close.
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