But Real did not wait long to make Berg pay, as Karim Benzema teed up Modric, who took a touch before curling a fine left-footed strike into the bottom corner.
Real captain Sergio Ramos grabbed his side's third goal with a powerful header from a corner before Al Ain, who shocked South American champions River Plate in the semifinals, got a late consolation goal through Japanese defender Tsukasa Shiotani.
Real could have won by more.
Japan protests against Russian military installations on Kuril Islands
Moscow , meanwhile, says it is perturbed by Japan's roll-out of the Aegis Ashore U.S. missile system. It was unclear whether the F-2 successor would be made-in-Japan or jointly developed.
It was a record fourth triumph in this competition for Real, with them surpassing LaLiga rivals Barcelona's tally of three wins, as boss Santiago Solari celebrated his first trophy in management.
Earlier, River beat Kashima Antlers 4-0 for third place.
Bale, scorer of a hat-trick in the semi-final victory over Kashima Antlers, nearly made it 2-0 in spectacular fashion shortly after the break, sending the ball just over with an overhead kick, and moments later a Benzema shot bounced wide. The game pits the heavily favored giant of Spanish soccer against the champion of UAE's Arabian Gulf League (and the Club World Cup's "home" team) for the right to crown itself king of the world. Gonzalo Martinez scored twice on what is expected to be his final appearance before joining MLS champions Atlanta United.