Home skipper Kane Williamson (70) and Colin Munro (56) had seemingly set New Zealand on course to reach the 172-run victory target before their chase stalled and the hosts were dismissed for 155 in exactly 20 overs.
After being asked to bat, former captain Mohammad Hafeez (61) helped the tourists overcome a slow start, hitting Trent Boult for back-to-back fours and clobbering Corey Anderson for two boundaries and a six in the same over.
Pakistan skipper Shahid Afridi (23) hit two sixes and a couple of fours in the same Matt Henry over in a typically busy eight-ball knock, while Umar Akmal (24) chipped in with a 14-ball cameo to help Pakistan put up a competitive 171-8 total.
Amir received visa clearance for New Zealand only last week and appeared nervous with the new ball in his hand, starting with a wide and soon opting for a change of ends.
However, he came close to claiming a wicket in his second over but Afridi spilt a simple catch to reprieve Williamson, then on five.
Batting at the other end, Munro decided to counter-attack, belting three sixes and a four off the last four deliveries of Umar Gul's second over en route to a 25-ball fifty.
Pakistan regained control of the contest when Wahab Riaz (3-34) ended Munro's eventful knock before Afrid dismissed Grant Elliott and Luke Ronchi in the same over to jolt the hosts.
Amir endured a second slice of misfortune when Sohaib Maqsood dropped debutant Todd Astle off his bowling before he finally tasted success in the penultimate over, this time Afridi making no mistake in taking the catch to account for Henry.
Hamilton hosts the second of three Twenty20 Internationals between the sides on Sunday before they embark on a three-match one-day international series.
(Reporting by Amlan Chakraborty in New Delhi; Editing by John O'Brien)
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