Asked to bat in seam-friendly conditions at the Basin Reserve, the tourists slumped to 23-4 in the 10th over and struggled to recover, eventually dismissed in the 44th over.
Coming off a productive Test series against England, Henry was a handful with the new ball, taking the key wicket of opener Pathum Nissanka for nine in the fifth over.
The relentless seamer returned to help mop up the tail, finishing with 4-19 off 10 overs.
Seamers Nathan Smith and Jacob Duffy snared two wickets each, capitalising on bouncy conditions offering some sideways movement.
Opener Avishka Fernando rode out the early storm to top score with 56 off 63 balls, hitting six fours and a six.
He put on 87 for the fifth wicket with Janith Liyanage, who departed for 36, caught in the deep from off-spinner Mitchell Santner.
Captain Santner also took three catches and produced a sharp run-out of danger man Kamindu Mendis, hitting the stumps from side-on when a risky single was attempted.
Tailenders Wanindu Hasaranga (35) and Chamidu Wickramasinghe (22) scored late runs before both were caught off Henry.
Sri Lanka handed an international debut to 23-year-old seamer Eshan Malinga.
New Zealand omitted experienced wicketkeeper-batsman Tom Latham, who is experiencing "soreness".
New Zealand won the preceding Twenty20 series against Sri Lanka 2-1.
