Marin Cilic beat Lucas Pouille in straight sets to secure Croatia’s second-ever Davis Cup title.
Cilic beat Pouille 7-6 6-3 6-3 to give them an unassailable 3-1 lead over hosts and defending champions France.
France had taken the final into a third day when Nicolas Mahut and Pierre-Hugues Herbert won Saturday’s doubles rubber.
“It’s not every day that you become a world champion,” 30-year-old Cilic said.
“For us it’s a dream come true, for this nation. You can see the fans are so passionate and they are here enjoying themselves. In Croatia it’s going to be incredible too.”
This is the final staging of the team competition in its current format before it changes next year.
Croatia last won the competition in 2005.
Cilic’s victory on Sunday meant the scheduled fifth match between Jo-Wilfried Tsonga and Borna Coric was a dead rubber.
Coric won his other singles rubber against Jeremy Chardy, while Cilic put Croatia 2-0 up by beating Tsonga.