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.
Noah ‘disgusted’ at changes
Afterwards, France captain Yannick Noah, who led his nation to Davis Cup glory in 1991 and 1996 and again last year, criticised the changes to the event.
The current Davis Cup format is being replaced by an 18-nation World Cup-style event to be held in Madrid in November 2019, with the International Tennis Federation having agreed a $ 3bn (£2.34bn), 25-year partnership with Kosmos, the investment group founded by Barcelona defender Gerard Pique.
But a furious Noah, the last home player to win the men’s singles at the French Open in 1983, expressed his displeasure to ITF president David Haggerty.
“I said I’m disgusted and upset to his face. It’s the truth. It’s the way I feel,” Noah told reporters.
“Everyone has a right to feel differently but I feel I owe the Davis Cup because it means so much to me as a player, as a spectator.
“We have people who have decided it doesn’t matter, I don’t know if they don’t know, or they don’t care. But as I told the president I’m not from his world.
“When people tell us it’s still going to be the Davis Cup, they are lying.
“We talk about money but how much is it worth for the ball boy, who is from here, to shake the hand of Lucas Pouille and have a picture with him?
“How much is that worth? How much is the dream in dollars? This will never happen again.”