Atletico's Diego Simeone also took the coach of the season award while Barcelona's Uruguay striker Luis Suarez was named the top non-EU player.
Curiously, though, Barca's Argentina international Messi was recognised as the best forward at the award ceremony in Valencia yesterday.
Atletico dominated the awards as Jan Oblak was best goalkeeper, Diego Godin top defender and Frenchman Griezmann also claimed the fans' award.
Real Madrid weren't left empty-handed, though, as Luka Modric was named best midfielder.