The 2019 runners-up, who overcame Italy in a thrilling semi-final on Saturday, can claim the crown if world number six Felix Auger-Aliassime beats Alex de Minaur in the second match at the Palacio de deportes Martin Carpena.
If the tie goes into a deciding doubles rubber, Auger-Aliassime and Vasek Pospisil are scheduled to take on Max Purcell and Jordan Thompson.
Shapovalov fell short in a physically punishing three-setter against Lorenzo Sonego in the semi-final, but the 18th-ranked left-hander wasted little time establishing a 4-0 lead in the opening set before Kokkinakis got on the board.
Having wrapped up the set comfortably to put Canada on the front foot, Shapovalov broke in the third game of the second set and did not look back as he closed out the victory in 89 minutes.
Australia are looking to win the title for the first time since 2003, when captain Lleyton Hewitt was part of the team who defeated Spain in the final.
