Having taken five wickets in the first innings, Herath went one better in the second with figures of six for 43 off 18 overs, his haul coming at an overall cost of 108 runs.
It was the bedrock of the Sri Lanka victory as New Zealand continued their miserable form, underlined by their woeful second-innings effort of 118 in which Daniel Flynn top-scored with just 20.
Sri Lanka were given a run chase of just 93 for a third-day victory which was reached with ease with Tharanga Paranavitana and debutant Dimuth Karunaratne quickly polishing off the runs in 18.3 overs.
Karunaratne was seemingly spraying boundaries at will in an innings more reminiscent of Twenty20 cricket, his unbeaten run-a-ball 60 including nine fours.
Fittingly, his ninth boundary produced the winning runs as he swivelled at the crease and pulled his shot along the ground from Doug Bracewell's short delivery.
The Black Caps lost a two-Test series to India in the summer having already been beaten by the West Indies earlier in the year.
Paranavitana was more conservative than Karunaratne, making 31 off 51 balls with three fours and one six.