Shapovalov, ranked 18, won 6-2, 6-4 against his 95th-ranked opponent in one hour and 29 minutes in southern Spain to put Canada 1-0 against the 28-time winners.
The 18th-ranked Canadian had been out of sorts in a semi-final defeat but hit back against Kokkinakis with 23 winners to the Australian's five.
World number six Felix Auger-Aliassime will bid to give Canada their first Davis Cup title against 24th-ranked Alex De Minaur in the second singles rubber.
If De Minaur wins, the title will be decided in the doubles.
Shapovalov raced to a 4-0 lead after a quarter of an hour in Malaga, sealing the first set with his ninth winner after half an hour.
The Canadian staved off three break points in the fourth game of the second set.
Kokkinakis dug in to convert his first break point of four when Shapovalov was serving for the match and held serve but Shapovalov was not to be denied sealing victory on his second match point.
Canada were runners-up to Spain in their only previous final in 2019, while Australia haven't played in the final since 2003 when they claimed a 28th title.
