Tahir claimed five wickets as the Proteas skittled New Zealand for 107 with five overs to spare after batting first and scoring 185 for six in their 20 overs.
"It's a very special feeling," Tahir said after his man-of-the-match performance at Eden Park, only the third five-wicket haul by a South African in a T20 international.
New Zealand captain Kane Williamson conceded his team was outplayed and must improve with five one-dayers and three Tests against the Proteas looming.
"It's frustrating from our perspective, we didn't really fire a shot," he said.
Amla made the most of Eden Park's short boundaries to smash 62 off 43 balls, receiving support from skipper Faf du Plessis (36) and JP Duminy (29).
They looked set to make 200-plus but Trent Boult and Colin de Grandhomme put the brakes on with two wickets apiece.
Boult was the one standout for New Zealand, conceding only eight runs in his four overs.
But with dangerman Martin Guptill injured and Ross Taylor controversially omitted from the T20 side, the Black Caps woes with the bat were apparent.
Chris Morris made a perfect start with two wickets in his first two balls for South Africa, coaxing an edge of debutant Glen Mitchell then skittling Colin Munro with his next ball.
Phehlukwayo then took Williamson's wicket with a short delivery and dismissed Corey Anderson in his next over.
Not to be outdone, Tahir dispatched Tom Bruce (33) and Luke Ronchi (0) in succesive deliveries, before wrapping up the tail.
The first one-dayer will be played in Hamilton on Friday.
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