Fast bowler Brunt finished with three wickets for 31 as New Zealand posted 231 for eight in their 50 overs with opener Maria Fahey making 61 and all-rounder Sophie Devine 50, including two sixes.
England captain Charlotte Edwards all but saw the hosts to victory with 70 but when she was the eighth batsman out, in the 48th over, her side still needed eight to win.
And the match was thrown back into the balance when Holly Colvin was out in the next over before Brunt, who ended on nine not out, hit the winning runs with three balls to spare.
“I’m really pleased to get the series off to a winning start,” Edwards said. “Katherine put in a fantastic bowling performance and there were some good contributions from the batters.”
The second game of the five-match series takes place at Taunton on Monday.