The Buccaneers also signed defensive tackle Mitch Unrein to a contract.
Nick Folk was released after four games last season before Patrick Murray handled the kicking duties the rest of the way.
Catanzaro has made 103 of 122 field-goal attempts during his four NFL seasons. He spent his first three with the Arizona Cardinals, with whom he set a franchise record by making 53 extra points in 2015.
The 27-year-old spent last season with the New York Jets, going 25 of 30 on field-goal attempts.
Catanzaro played college ball at Clemson, where he was a three-time All-ACC selection and a two-time semifinalist for the Lou Groza Award.
The 6-foot-4, 301-pound Unrein had 32 tackles and 2.5 sacks last season with the Bears.