A draw appeared the most likely result after rain washed out Friday's play, but Kyle Jamieson removed home captain Joe Root with the opening delivery of the day for 42 before the outstanding Southee (6-43) put the tourists on top.
Having trapped Ollie Pope lbw for 22, the seam bowler removed both Dan Lawrence and debutant James Bracey for ducks, leaving England at 140-6 and in danger of being asked to follow on.
That possibility disappeared thanks to a seventh-wicket stand worth 63 between Burns – who had witnessed the carnage unfold at the non-striker’s end – and Ollie Robinson, the latter following up four wickets with the ball by making a crucial 42.
Southee eventually ended the partnership and with Mark Wood (0) and Stuart Broad (10) also departing before he had reached three figures, Burns had to rely on last man James Anderson to keep him company through to three figures.
The left-handed opener was given two reprieves along the way, wicketkeeper BJ Watling failed to take a stumping opportunity when on 77, while Southee shelled a catch at second slip 11 runs later.
Burns' third Test hundred eventually arrived from the 267th delivery faced, and he had continued on to 132 by the time Southee had him caught behind, both players knowing they had done enough to see their names added to the famous honours board. England were dismissed for 275, meaning they trailed by 103.
New Zealand extended that advantage to 165 as they closed on 62-2, losing Devon Conway (23), who made a double century in the first innings, as well as Kane Williamson (1) along the way, with Robinson (2-8) removing both.
A well deserved reception for our centurion!June 5, 2021
All seams good for Black Caps
Southee claimed his 12th five-wicket haul in Tests and his second at Lord's, having also taken 6-50 at the famous venue eight years earlier. Richard Hadlee (six) and Chris Cairns (three) are the only Kiwi bowlers to have taken five in an innings on more occasions in England.
Burns props up England
Left out during the tour of India, Burns made sure he seized his opportunity at the top of the order. There were a few bumps along the way – he was twice hit on the helmet by short balls – and the odd moment of good fortune, but a knock that included 16 fours and a solitary six was pivotal for his side.