They beat Fijian team Marama Vou 68-41 with shooter Chelsea Lewis named Player of the Match.
The Welsh team lost 65-41 to the host nation's Pulse Netball in Nelson in their opening game.
Kiwi club SkyCity Mystics are Celtic Flames' next opponents in Pool B. They face each other on Wednesday.
The Welsh team brought in two New Zealand international players for the tournament with Anna Thompson, 31, and 22-year-old Temalisi Fakahokotau joining 10 Wales players in the squad.
Teams from Australia, South Africa, and Trinidad and Tobago make up the eight-team competition.
Celtic Flames included eight of the Celtic Dragons line-up from the UK Superleague, plus former Dragons Chelsea Lewis and Nia Jones.
Fakahokotau normally plays either goal keeper or goal defence, while Thompson covers all three attacking roles.
Welsh netball side Celtic Flames have earned their first win in the inaugural International Super Cup competition in New Zealand.