Tavares, 21, has made 25 league appearances for Benfica since coming through the club's youth academy and making his senior debut in the Portuguese Super Cup win over Sporting Lisbon in August 2019.
"He is a young player with great promise who has developed very well with Benfica in recent seasons and has also shown his quality by being part of Portugal Under-21s," Arsenal manager Mikel Arteta said in a statement.
"Nuno's arrival will give the squad extra strength and options in defence, particularly with this energy on the left side of the pitch."
Arsenal finished eighth in the Premier League last season and will begin their 2021-22 campaign at newly promoted Brentford on Aug. 13.