The 32-year-old came out on top in a three-man shortlist that included teammate Manuel Neuer and Manchester City’s Kevin De Bruyne.
Lewandowski’s goals helped the Bavarians to their sixth European crown, beating Paris Saint Germain in the final in Lisbon.
Lewandowski ended the season as top scorer in the competition with 15 goals to his name. The Poland international scored 55 goals in 47 appearances across all competitions for Bayern Munich last season.
"I have to say it's amazing," Lewandowski said in his acceptance speech.
"It's amazing, if you work so hard and you get this trophy, it's something special.
"I have to say thank you to my team-mates, coaches, all the staff because they work very hard every day to prepare me for the games and also to my family because they support me a lot.
"I've always dreamed of playing in the biggest stadiums at the biggest club in the world. I'm very grateful and very proud and very happy."
Lewandowski also won the forward of the year while teammates Neuer and Joshua Kimmich were named goalkeeper and defender of the year respectively. De Bruyne won midfielder of the year and Bayern Munich manager Hansi Flick picked up coach of the year.
Liverpool’s Virgil Van Dijk won the UEFA Men’s player of the year in the 2018/2019 season, beating Messi and Ronaldo to the award after helping the Reds to their sixth European crown.
