Chasing 170 for victory, England reached the target with four balls to spare at the Wankhede Stadium, thanks to Roy's 55 off 36 balls at the top and some hard hitting by Jos Buttler and Ben Stokes in the end.
Roy, who struck seven fours and two sixes during his innings, got adequate support from Alex Hales (44) with the duo adding 77 for the opening wicket.
Kiwi spinner Mitchall Santer, who went for 16 in his first over, made a good comeback by dismissing Roy in the 9th over. Santner then scalped Joe Root for 12 to lead the fight-back for the Kiwis.
England needed 30 off the last three overs and Strokes (24) and Butler (8) kept the pressure on the Kiwi bowlers to guide England home.
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For New Zealand, spinners Santer and McCulllum took two wickets each.
Earlier, electing to bat, Kiwis opener Martin Guptill fell cheaply for 16, before skipper Kane Williamson anchored the innings with a knock of 63 to help the side put 169 for 8.
The Kiwi team, who initially blazed their way to 78/1 after the first seven overs, struggled after spin was introduced from both the ends.
Williamson, who initially hit all the bowlers for runs, was stumped by Butler off Rashid. He hit nine eloquent boundaries and a long six.
Ross Taylor who remained not out on 19, failed to get support from the other end. Taylor and Tim Southee did some big hitting in the last over. Southee remained unbeaten on 11.
For England, Liam Plunkett was the most expensive bowler as he gave 19 runs in his only over, while Chris Jordan, Stokes and Moeen Ali snapped a wicket each.
Brief Scores:
New Zealand: 169/8 (Kane Williamson 63, Ross Taylor 19 not out; Adil Rashid 3/15, R Topley 2/39) lost to England 170-4 (J Roy 55, Alex Hales 44; M Santer 2/24, N McCullum 2/25).