Opener Zaman (56) and captain Azam (66) stitched together a partnership of 78 runs, with Rizwan then scoring an unbeaten 77 off 86 balls to guide Pakistan over the finish line and help them take a 1-0 lead in the three-match one-day international series.
Michael Bracewell was the pick of the bowlers for New Zealand, dismissing Zaman and Imam-ul-Haq, with Tim Southee and Glenn Phillips also taking a wicket apiece.
Earlier on Monday, 19-year-old Naseem picked up his second five-wicket haul in international cricket to help Pakistan restrict New Zealand to 255 for nine after putting their opponents in to bat at Karachi's National Stadium.
"A good performance is fun only when the team wins. Everyone bowled well. It was reversing, and I just bowled to my strength," Shah said.
None of New Zealand's batters were able to build on their starts, with Tom Latham (42) and Bracewell (43) leading the scoring for the visitors.
Spinner Usama Mir had a debut to remember, taking the wickets of Kane Williamson and Latham.
The second ODI of the series will be held at the same venue on Wednesday.See all the statistics
