The 32-year-old Lewandowski, who was named the 2019-20 UEFA Men's Player of the Year in October, joins Luka Modric (2018) as the only players other than Messi (6) and Ronaldo (5) to win the FIFA men's award since 2008.
"To win such an award and share this title with [previous winners] Messi and Ronaldo is unbelievable and means so much to me," Lewandowski said. "I am very proud and happy. This is a great day for me, and also for my club and colleagues."
Manchester City defender Lucy Bronze won The Best FIFA women's award. She becomes the first player -- male or female -- from England to earn the honor.
Bronze edged Denmark's Pernille Harder and France's Wendie Renard to claim the FIFA award at a virtual ceremony in Zurich, Switzerland.
"I don't have the words right now to explain how I'm feeling," Bronze said. "If there's anything 2020 has taught us, [it's] to appreciate every moment you've got, not look too far ahead and live in the here and now.
"I appreciate it more than ever. I'll remember this moment forever, for the rest of my life."
Meanwhile, Bayern Munich's Manuel Neuer won the award for best goalkeeper, becoming the first German goalie to claim the prize. Son Heung-Min was given the Puskas Award for the best goal of 2020.
Liverpool manager Jurgen Klopp was named Coach of the Year for the second time. Netherlands manager Sarina Wiegman was awarded the women's Coach of the Year trophy.