Dong Hyeon Yoon got the first, having taken the penalty corner, before sprinting in on goal to deflect a drive past keeper Kyle Pontifex.
Jihun Yang got the second and third with slick, powerful drag flicks.
However that third goal acted as a spur to set up New Zealand's best period of the match.
Kane Russell punched a firm, low drive into the Korean goal from New Zealand's first penalty corner and it seemed to give the Black Sticks more self belief and confidence in their link play
Nick Wilson got the second, bundling the ball in from close range after combining well with captain Simon Child, who was celebrating his 250th international tonight.
Then came Hilton's late strike, set up by Nick Ross's drive into the circle.
''We showed really good fight in the second half and got the reward we deserved in the end," Child said.
''They looked after the ball really well (in the first half), had good structure and we were probably guilty of not taking our chances in the first half, but were more creative in the second."
The teams meet in game two at the same venue tomorrow night.
New Zealand are coming off a 2-1 series win over 13th-ranked Malaysia last week. This series continues their buildup to Rio.