Burns was dropped twice the previous evening and endured a couple of challenging moments on Sunday morning, but an unbroken 118-run stand alongside Root took England to 142 for two in response to New Zealand's 375 all out.
Root overturned a caught behind decision against him on 47 before going to lunch on 50 not out, with Burns unbeaten on 76, as England set their stall out to bat big.
England seamer Stuart Broad had said on Saturday night that "if you were good at batting, you'd fancy" the flat surface at Seddon Park, where he tipped Burns to capitalise on his moments of fortune.
New Zealand had scented blood after snaring Dom Sibley and Joe Denly, while Burns was shelled on 10 and 19 as England closed on 39 for two but the Black Caps bowlers lacked a cutting edge at the start of the third day.
Burns got himself into a brief tangle when a shorter delivery from Neil Wagner thudded into his chest while the England opener turned the next ball just out of reach of the diving leg gully fielder.
However, in Wagner's next over, Burns authoritatively pulled New Zealand's left-arm enforcer for back-to-back fours.
Wagner responded by beating the outside edge while Burns top-edged a pull short of the deep square-leg fielder before completing a nuggety half-century.
Root was looking much more composed at the other end, initially sprightly before slowing up a touch when Mitch Santner came into the attack.
But there was little troubling him until he was given out, apparently strangled down the leg-side off Daryl Mitchell, three shy of his 50.
Root immediately reviewed and the decision was overturned when technology showed he had not feathered through to the wicketkeeper, and the Yorkshireman reached a 145-ball 50 shortly afterwards.