Milne took five wickets for just 26 runs as Sri Lanka’s batters collapsed in Dunedin.
New Zealand are looking to level the series after Sri Lanka won the opening T20 match last Sunday in Auckland, sealing victory in a nail-biting Super Over.
Sri Lanka made a good start Wednesday as Kusal Perera (35) and Dhananjaya de Silva (37) put on a middle-order partnership of 62 off 46 balls to steer their side to 99-4 after 12 overs.
The tourists then lost their next six wickets for just 42 runs as New Zealand’s seamers found their length to devastating effect.
Charith Asalanka was the last Sri Lankan batter to offer any resistance, chalking up 24 runs off 19 balls.
Milne did the most damage for New Zealand, picking off Sri Lanka’s tailenders Dilshan Madushanka and Pramod Madushan without a run being scored. The three-match T20 series concludes in Queenstown on Saturday.
Sri Lanka have lost both the Test and one-day international series 2-0 during their tour of New Zealand.
