Daria Gavrilova had earlier defeated Elina Svitolina 6-4 7-6 (8-6) in the women's rubber of the mixed event.
Svitolina had six set points at 6-1 in the second-set tie-break but lost the last seven points of the match.
Kyrgios, ranked 30th in the world, beat Britain's world number two Andy Murray in the group stage on Wednesday.
The tournament hosts, who had two squads competing in the eight-team event, last won the competition in 1999.
Nick Kyrgios secured the Hopman Cup for Australia Green by beating Alexandr Dolgopolov 6-3 6-4 to give his team a 2-0 victory over Ukraine in Perth.