Sri Lanka had begun the day on 293 for five, still 10 runs in arrears overall.
Seamer Tim Southee took three wickets, including key man, captain Angelo Mathews cheaply.
Last pair Shaminda Eranga and Surenga Lakmal put on 59, with Eranga making a career-high 45 not out.
Southee and Trent Boult picked up four wickets apiece to share 13 in the match.
The second test starts at the Basin Reserve on Saturday.
Full scoreboard
New Zealand 1st innings
T LATHAM c Kaushal b Eranga 27
H RUTHERFORD b Lakmal 18
K WILLIAMSON b Prasad 54
R TAYLOR run out (Silva) 7
B McCULLUM c Karunaratne b Kaushal 195
J NEESHAM c Sangakkara b Mathews 85
B WATLING lbw Mathews 26
M CRAIG not out 12
T SOUTHEE c Thirimanne b Mathews 0
N WAGNER c Kaushal b Lakmal 4
T BOULT c Jayawardene b Lakmal 0
Sundries (4lb 7nb 2w) 13All out for 441Fall: 37 (Rutherford), 60 (Latham), 88 (Taylor), 214 (Williamson), 367 (McCullum), 420 (Neesham), 429 (Watling), 431 (Southee), 440 (Wagner), 441 (Boult).Bowling: S Lakmal 19.5-3-90-3 (2nb), S Eranga 18-1-82-1, A Mathews 12-2-39-3, D Prasad 12-2-62-1 (2w), T Kaushal 22-0-159-1 (5nb), L Thirimanne 2-0-5-0.Overs: 85.5
Sri Lanka 1st innings
D KARUNARATNE lbw Boult 0
K SILVA lbw Boult 4
K SANGAKKARA c Southee b Boult 6
L THIRIMANNE c Craig b Southee 24
A MATHEWS c Latham b Wagner 50
N DICKWELLA c McCullum b Southee 2
P JAYAWARDENE c Williamson b Wagner 10
D PRASAD c McCullum b Neesham 18
T KAUSHAL c Williamson b Wagner 6
S ERANGA not out 10
S LAKMAL c McCullum b Neesham 2
Sundries (3lb 1nb 2w) 6All out for 138Fall: 0 (Karunaratne), 8 (Silva), 15 (Sangakkara), 58 (Thirimanne), 60 (Dickwella), 88 (Jayawardene), 105 (Mathews), 118 (Kaushal), 128 (Prasad), 138 (Lakmal).Bowling: T Boult 11-4-25-3 (1w), T Southee 12-4-17-2, J Neesham 6.4-1-28-2 (1w), N Wagner 11-0-60-3 (1nb), M Craig 2-0-5-0.Overs: 42.4
Sri Lanka 2nd innings
D KARUNARATNE b Boult 152
K SILVA c Watling b Southee 33
K SANGAKKARA c Watling b Boult 1
L THIRIMANNE c Watling b Neesham 25
A MATHEWS c Watling b Southee 66
N DICKWELLA c Neesham b Boult 4
T KAUSHAL c Craig b Southee 12
P JAYAWARDENE c Southee b Craig 23
D PRASAD c Taylor b Southee 4
S ERANGA not out 45
S LAKMAL c Southee b Boult 16
Sundries (4b 20lb 1nb 1w) 26All out for 407Fall: 85 (Silva), 94 (Sangakkara), 181 (Thirimanne), 277 (Karunaratne), 287 (Dickwella), 307 (Kaushal), 320 (Mathews), 325 (Prasad), 348 (Jayawardene), 407 (Lakmal).Bowling: T Boult 39-8-100-4 (1w), T Southee 37-8-91-4, N Wagner 30-6-76-0 (1nb), M Craig 38-10-83-1, J Neesham 8-2-29-1, B McCullum 1-0-3-0, K Williamson 1-0-1-0.Overs: 154
New Zealand 2nd innings
T LATHAM c Mathews b Kaushal 17
H RUTHERFORD c Dickwella b Eranga 10
K WILLIAMSON not out 31
R TAYLOR not out 39
Sundries (6lb 2nb 2w) 10Two wickets for 107Fall: 23 (Latham), 43 (Rutherford).Bowling: S Lakmal 6-2-16-0 (1w 1nb), T Kaushal 13-0-48-1 (1nb), S Eranga 7-2-20-1, D Prasad 4.4-1-17-0 (1w).Overs: 30.4Toss: Sri LankaUmpires: Bruce Oxenford, Richard Illingworth
TV umpire: Steve Davis
Match referee: Chris Broad
Series: New Zealand leads the series 1
- additional reporting AAP