Herath claimed 6-43 in the second innings as New Zealand collapsed on Monday morning, making just 118 all out, having slumped from 35-1 to 79-7 in another disappointing batting display.
Only one batsman could get to 20, despite eight players making double figures, as Herath added to his figures of 5-65 in the first innings.
Sri Lanka wasted no time in chasing down its victory target of 93, with Dimuth Karunaratne (60 not out) and Tharanga Paranavitana (31 not out) putting on an impressive undefeated stand in 18.3 overs.
The teams now head to Colombo for the second Test.
