France had given themselves a lifeline at 2-1 after Nicolas Mahut and Pierre-Hugues Herbert’s victory in Saturday’s doubles but Cilic snuffed out any hopes of a comeback with a 7-6 6-3 6-3 win to give his side an unassailable 3-1 lead.
“It’s not every day that you become a world champion,” said the 30-year-old in his post-match interview after guiding his country to their first Davis Cup triumph since 2005.
“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.”
Cilic’s win meant the scheduled fifth match between Jo-Wilfried Tsonga and Borna Coric was a dead rubber and therefore not played.
This was the final staging of the team competition in its current format before it changes next year.