"My father always pushed me to fight for my dreams, I owe him a lot for everything," said Modric, whose silky midfield skills helped Real to a third straight European crown and then propelled Croatia to the World Cup final.
"It's an incredible moment for me, I feel really excited and proud of this great award," said the 32-year-old, who started out at Dinamo Zagreb and then had four years with Tottenham before Real prised him away six years ago.
Modric, whose success sets him up as a favourite to win the Ballon d'Or world player of the year accolade, saw off former Real teammate Cristiano Ronaldo and Liverpool's Egyptian striker Mohamed Salah for the overall award.
Real Madrid's achievement in winning the Champions League for the third season running was further recognised as Modric also won the UEFA award for the best midfielder of last season.
His clubmate Sergio Ramos was named best defender while Real shot stopper Keylor Navas won the best goalkeeper prize.
Ronaldo, who did not appear at the ceremony, was awarded the award for best striker chosen by Champions League and Europa League coaches from last season and journalists.
The award "recognises the best players, irrespective of nationality, playing for a club within the territory of a UEFA member association during the past season. Players are judged on their performances in all competitions – domestic, continental and international".
Danish international Pernille Harder, of German club Wolfsburg, won the UEFA women's player of the year prize.
Wolfsburg lost to dominant French outfit Lyon in the Champions League final in Kiev in May.
There was also an award for David Beckham, the former Manchester United, Real Madrid and England star collecting the UEFA President's Award for his achievements in the game and his work since retiring as a player.