The Italian brothers rallied to beat co-runners-up Sweden (69), the defending champion, and Ireland (70), the leader after each of the first three rounds.
Three straight birdie putts on the back nine helped secure Italy's first title, despite each of the top three teams owning a share of the lead during the final round.
"It (was) a tough day and we have been playing against some of the best golfers in the world," Francesco Molinari said after a winning total of 29-under 259 to 28-under 260 for Ireland and Sweden. "I think we deserved it as we attacked from the first day with every putt and every shot. It's really great for Italy."
"It's a pity we were not able to pull it off," said Ireland's Rory McIlroy.