Coric's victory followed Donna Vekic's (27) comfortable 6-3, 6-4 win over Celine Naef (18) and means Croatia become the first team to lift the trophy following the tournament's return to the tennis calendar after a four-year absence.
Switzerland are among the most successful countries in the event - which began in 1989 and was named after Australian great Harry Hopman - having won four titles including the 2019 crown on the back of the now-retired Roger Federer's heroics.
Without the seasoned Belinda Bencic due to withdrawal, the young Swiss team scored victories over Denmark and France to top Group A this year but ran out of steam against Croatia in the title clash on the claycourts of Nice.
World number 15 Coric raced through the opening set in 31 minutes and recovered from an early loss of serve in the next to close out the win and break Swiss hearts.
Earlier, Vekic exchanged breaks with Naef early in the first set of the opening match of the tie, before the more experienced world number 22 pounced on her Swiss opponent's serve in the eighth game en route to wrapping up the set.
Vekic showed plenty of patience and skill in the second set to stave off a comeback attempt by Naef, who slipped and took a tumble late in the contest before crashing to her third defeat in as many singles matches in the tournament.
Croatia previously won the title in 1996 when they beat the same opponents in the final in Perth on a hardcourt, with Goran Ivanisevic and Iva Majoli the architects of that triumph.
