They bounced back from conceding an early goal to David Condon and were 2-1 up with strikes from Kane Russell and Hayden Phillips, with a classy backhand shot. Barry Middleton got Britain's second goal five minutes before halftime.
Both teams had late opportunities but smart saves by keepers Devon Manchester (NZ) and George Pinner kept the scores locked up.
New Zealand had lost their opening game 2-1 to world No 1 Australia and next face 11th-ranked Spain early on Wednesday (NZT).
Earlier the women's Black Sticks got their campaign off to a strong start with a 4-1 win over South Korea.
Goals from Kirsten Pearce, Charlotte Harrison, Gemma Flynn and Petrea Webster ensured the fourth-ranked women made a strong early statement. They next play Germany tomorrow.
