Cantabrians Tom Latham (176) and Henry Nicholls (162no) found their form from the first test scoring massive hundreds to help the Blackcaps reach 585/4d.
Captain Kane Williamson declared with just under an hour to go on the days play giving Boult and Southee 14 overs to bowl at the visitors.
Colin de Grandomme chipped in, scoring the fastest test fifty by a New Zealander off just 28 balls.
Boult struck first, claiming Dinuth Karunaratne caught down the leg side in the first over.
Southee dismissed other opener Danushka Gunathilaka the next over claiming him caught behind.
Sri Lanka ended the day 24/2.
The hosts will need just eight wickets in the next two days to claim a record fourth straight test series win.
On day one of the test, Latham only managed 10 runs as New Zealand were bowled out for 178, before Trent Boult took six wickets with the ball to dismiss Sri Lanka for 108 in their innings.
The 26-year-old scored an outstanding 264 not out in the Blackcaps’ only innings in the first test in Wellington which finished in a draw after day the final day was washed out.
In reaching his 200, Latham became the sixth New Zealand batsman to achieve the milestone on the Basin Reserve, behind Bendon McCullum, Martin Crowe, Kane Williamson, Lou Vincent and Mathew Sinclair. (Courtesy Newshub New Zealand)