Lewandowski, 32, joins Luka Modric (2018) as the only players other than Lionel Messi (6) and Cristiano Ronaldo (5) to have won the FIFA Men's Player of the Year since 2008.
He also led Poland to Euro 2020 qualification and was named the 2019-20 UEFA Men's Player of the Year in October.
"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.
"This award also belongs to my colleagues, the coach and Bayern in general. It is an incredible feeling, a lot of emotions."
Manchester City and England defender Lucy Bronze won The Best FIFA Women's Player of the Year. She is the first player, male or female, from England to land the honour from FIFA.
"I don't have the words right now to explain how I'm feeling," Bronze said. "If there's anything 2020 has taught us 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."
Bayern's Manuel Neuer won the award for best goalkeeper -- becoming the first German goalie to take home the prize -- while Son Heung-Min was handed the Puskas Award for the best goal with his stunning end-to-end run and strike in Tottenham's 5-0 win over Burnley on Dec. 7, 2019.
Liverpool boss Jurgen Klopp was named Coach of the Year for the second time. He also won it in 2019, making him the first man to win the award twice. Netherlands manager Sarina Wiegman was given the women's Coach of the Year award, also for the second time, joining Jill Ellis (two) and Silvia Neid (three) as the only multi-time female winners.
FIFA's The Best award is the rebirth of the FIFA World Player of the Year award, which ran parallel to the Ballon d'Or award from 1991 to 2009. The two awards then merged to create the FIFA Ballon d'Or from 2010 to 2015 before FIFA The Best was reintroduced in 2016.
The winners, who were voted on by selected members of the media, national team captains and coaches as well as fans, were announced via a virtual ceremony on Thursday.
Lewandowski, whose 55 goals helped Bayern win the Treble in 2019-20, said he was disappointed the Ballon d'Or award had been cancelled due to COVID-19. But he added his first FIFA The Best award, edging Ronaldo, who has won it twice, and Messi, who won in 2019.
Ronaldo scored 31 goals in Serie A last term as Juventus won the title, while Messi hit 25 in La Liga. Lewandowski scored 34 in the Bundesliga to finish second in the European Golden Shoe list.
Although Ronaldo and Messi missed out on the top prize, they were both named alongside Lewandowski in the FIFA FIFPro World XI. Bayern Munich and Canada international Alphonso Davies also made the XI, becoming the first player from his country to make the select group.
U.S. women's forward Megan Rapinoe was picked for the FIFA FIFPro World Women's XI, as was Bronze, Tobin Heath of the U.S. and France's Wendie Renard.
Rapinoe tweeted a statement after the announcement, saying she was honoured, but didn't think she deserved to be in the team after not playing in a match since March.
FIFPro World XI's Men's Team
Alisson Becker (Liverpool and Brazil), Trent Alexander-Arnold (Liverpool and England), Virgil van Dijk (Liverpool and Netherlands), Sergio Ramos (Real Madrid and Spain), Alphonso Davies (Bayern Munich and Canada), Joshua Kimmich (Bayern Munich and Germany), Kevin de Bruyne (Manchester City and Belgium), Thiago Alcantara (Liverpool and Spain), Lionel Messi (Barcelona and Argentina), Robert Lewandowski (Bayern Munich and Poland), Cristiano Ronaldo (Juventus and Portugal)
FIFPro World XI's Women's Team
Christiane Endler (PSG and Chile), Lucy Bronze (Manchester City and England), Wendie Renard (Lyon and France), Millie Bright (Chelsea and England), Delphine Cascarino (Lyon and France), Barbara Bonansea (Juventus and Italy), Veronica Boquete (AC Milan and Spain), Megan Rapinoe (OL Reign and USA), Pernille Harder (Chelsea and Denmark), Vivianne Miedema (Arsenal and Netherlands), Tobin Heath (Manchester United and USA)
Information from Reuters was used in this story.