Cardinals rank second only to the pope in the Church hierarchy and serve as his closest advisers
Man of the match Iyer hit the winning runs with a booming six to be unbeaten on 58 as India reached 204 for four in reply to New Zealand's 203 for five.
Opener Rahul (56) set up the run chase in a 99-run stand off 50 balls for the second wicket with captain Virat Kohli (45).
Although the tourists only arrived in New Zealand on Tuesday, Kohli said they were determined that would not be a factor in how they approached the game.
"We never spoke of the jet lag inside the team. We thought we're not going to take anything for an excuse, we're just going to come out there and back our skills," Kohli said.
"We were just looking forward to going on the park and that's key to doing well at the international level and probably that's the reason why we put in a performance like that today."
A 200-plus score on most grounds would usually be a formidable target in Twenty20 cricket but it is not so imposing at Auckland's Eden Park with its short boundaries.
New Zealand also did not help their defence of 203 by bowling 10 wides and missing wicket opportunities with sloppy fielding.
Kohli won the first play of the day when he called correctly at the toss and elected to bowl.
Half-centuries by Colin Munro (59), Kane Williamson (51) and Ross Taylor (54 not out) formed the basis of the New Zealand innings.
When India started their run chase, New Zealand took the bold option of throwing the new ball to spinner Mitchell Santner for the second over and after conceding 12 off his first three deliveries the left-armer claimed the wicket of Rohit Sharma for seven.
But it was merely a momentary blip as Kohli joined Rahul to pound the New Zealand attack.
At 59 for one in the sixth over New Zealand butchered a double chance to remove Rahul on 27 when he was slow away on a quick single.
He was so far out of his crease that New Zealand had time for two throws at the stumps and both missed.
Kohli on 33 skied a Blair Tickner delivery which Ish Sodhi had plenty of time to get under but he missed the ball completely.
But when the pair had put on 99 off 50 deliveries New Zealand briefly fought back as the catches stuck.
Tim Southee caught Rahul and Shivam Dube in the deep and a diving Martin Guptill accounted for Kohli (45) as India slumped from one for 99 to four for 142 in the 14th over.
But Iyer and Manish Pandey recovered the momentum for the tourists and knocked off the remaining 62 runs needed for victory in 34 balls with the match finishing on Iyer's third six.
The second match in the series is in Auckland on Sunday.Scoreboard
New Zealand
M. Guptill c Sharma b Dube 30
C. Munro c Chahal b Thakur 59
K. Williamson c Kohli b Chahal 51
C. de Grandhomme c Dube b Jadeja 0
R. Taylor not out 54
T. Seifert c Iyer b Bumrah 1
M. Santner not out 2
Extras (lb1, w5) 6
Total (5 wkts, 20 overs) 203
Did not bat: Ish Sodhi, Tim Southee, Blair Tickner, Hamish Bennett
Fall of wickets: 1-80, 2-116, 3-117, 4-178, 5-181
Bowling: Bumrah 4-0-31-1 (2w), Thakur 3-0-44-1, Shami 4-0-53-0 (3w), Chahal 4-0-32-1, Dube 3-0-24-1, Jadeja 2-0-18-1
India
R. Sharma c Taylor b Santner 7
K. Rahul c Southee b Sodhi 56
V. Kohli c Guptill b Tickner 45
S. Iyer not out 58
S. Dube c Southee by Sodhi 13
M. Pandey not out 14
Extras (10w, nb1) 11
Total (four wkts, 19 overs) 204
Did not bat: Ravindra Jadeja, Shardul Thakur, Yuzvendra Chahal, Mohammed Shami, Jasprit Bumrah
Fall of wickets: 1-16, 2-115, 3-121, 4-142
Bowling: Southee 4-0-48-0, Santner 4-0-50-1 (3w), Bennett 4-0-36-0 (1w), Tickner 3-0-34-1 (1w, 1nb)), Sodhi 4-0-36-2 (5w)
Toss: India
Result: India won by six wickets
Umpires: Chris Brown (NZL), Shaun Haig (NZL)
TV Umpire: Ashley Mehrotra (NZL)
Match referee: Chris Broad (ENG)
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