The Atletico Madrid striker, who also provided two assists, scored both of France's goals in their 2-0 semi-final win over Germany and winners against Albania in the group stage and Ireland in the Round of 16.
Four Portuguese players and three from Germany were named in the team of the tournament.
Portugal captain Cristiano Ronaldo was joined by Pepe, Raphael Guerreiro and goalkeeper Rui Patricio.
Germany were represented by defenders Joshua Kimmich and Jerome Boateng and midfielder Toni Kroos.
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Wales captain Gareth Bale, who inspired his country to a historic semi-final against Portugal and is Ronaldo's club mate at Real Madrid, was overlooked but team mates Joe Allen and Aaron Ramsey were included.
France's Dmitri Payet and Griezmann also made the list.
Team of the tournament:
Goalkeeper: Rui Patricio (Portugal)
Defenders: Joshua Kimmich (Germany), Jerome Boateng (Germany), Pepe (Portugal), Raphael Guerreiro (Portugal)
Midfielders: Toni Kroos (Germany), Joe Allen (Wales); Antoine Griezmann (France), Aaron Ramsey (Wales), Dmitri Payet (France)
Forward: Cristiano Ronaldo (Portugal)
(Reporting by Ian Rodricks in Bengaluru, editing by Alan Baldwin)
