Man-of-the-match Fletcher hit one four and four sixes while left-hander Bravo cracked two sixes.
The pace pair of Tim Southee (2-20) and Trent Boult (2-28) were the mainstays of New Zealand's effort after they won the toss and chose to bowl.
Bolt trapped Lendl Simmons (1) leg before wicket with a yorker before Southee removed Dwayne Smith (8) at 10-2.
Fletcher and Bravo enjoyed their stand either side of an hour and a half of rain interruption.
But once both departed, the innings lost momentum, with allrounder Corey Anderson (2-32) also picking up two wickets.
New Zealand's chase lost Jimmy Neesham early when he dragged on a yorker from Krishmar Santokie at 12-1.
But McCullum shared a second-wicket stand of 53 with Kane Williamson (19) to engineer his side's victory.
Sammy struck three times during the light drizzle but New Zealand was always ahead on the rain rule.
The second and final match in the series is at the same venue on Monday.
- AP