The home side need another 106 runs to avoid the follow-on.
Southee, who has been in good form recently after he took 4-46 in the first innings of the first test last week, went to lunch with figures of 4-35 next to his name.
Scoreboard
Scoreboard at lunch on day three of the second cricket test between New Zealand and Sri Lanka being played at the P Saravanamuttu Stadium in Colombo today:
New Zealand
First innings 412
Sri Lanka
First innings (overnight: 43-3)
T Paranavitana c van Wyk b Southee 40
T Dilshan b Southee 5
K Sangakkara c Boult b Southee 0
M Jayawardene c Williamson b Boult 4
A Mathews c Guptill b Southee 47
T Samaraweera not out 0
P Jayawardene not out 4
Extras (3lb, 1w, 3nb) 7
Total (for 5 wickets, 43.0 overs) 107
Fall: 1-7 (Dilshan), 2-7 (Sangakkara), 3-12 (Jayawardene), 4-102 (Paranavitana), 5-103 (Mathews)
Bowling: T Southee 14-2-35-4 (1nb), T Boult 13-4-26-1 (1w, 2nb), J Patel 9-3-16-0, T Astle 2-0-6-0, D Bracewell 5-0-21-0