Modric won the Champions League with Real and then played a major role in Croatia's run to the World Cup final where they were beaten by France.
"This award is not just mine. It is my team mates' from Real Madrid and Croatia. Without my coaches, I would not have won this and without my family I would not be the player I am today," said 33-year-old Modric.
It is the first time since Brazilian Kaka's triumph in 2007 that the men's prize was not won by either Cristiano Ronaldo or Barcelona's Argentina forward Lionel Messi -- neither of whom attended the ceremony at London's Royal Festival Hall.
Portugal international Ronaldo, who joined Juventus from Real Madrid in the close season, was second in this year's awards with Liverpool's Mohamed Salah in third place.
The 32-year-old Marta, whose Orlando Pride team reached the U.S. National Women's Soccer League (NWSL) playoff semi-finals, had won the previous FIFA Women's Player of the Year award five times, most recently in 2010.
The Brazilian, who led her country to the Copa America title in April, beat Olympique Lyonnais pair Ada Hegerberg of Norway and Dzsenifer Marozsan of Germany to the award.
France's World Cup-winning manager Didier Deschamps took the Men's Coach of the Year award, with compatriot Reynald Pedros claiming the women's award after winning the Champions League with Lyon.
Voting for the player and coach categories was conducted by national team captains and managers, selected journalists and, for the first time, an online poll of fans.
Real Madrid's Belgium international Thibaut Courtois won the Goalkeeper of the Year Award.
Salah's goal for Liverpool against Merseyside rivals Everton in the Premier League won the Goal of the Year, which is voted for by fans, ahead of Gareth Bale's spectacular overhead kick for Real Madrid in the Champions League final.
Peru's supporters won the FIFA Fan Award and German Lennart Thy, who plays for Turkish club BB Erzurumspor, won the FIFA Fair Play Award after donating stem cells to help a leukaemia patient.
(Writing by Simon Evans; Editing by Ken Ferris)
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