The Browns waived Gonzalez in a corresponding move, the team announced Monday afternoon.
Joseph, who is from South Africa and played at Florida Atlantic, was in training camp with the Miami Dolphins this year and made all three of his attempts during the preseason. He was one of several kickers to try out for the Browns on Monday.
A seventh-round pick in 2017, Gonzalez made 15 of 20 field goals last season but missed two attempts as well as two extra points in the 21-18 loss to the New Orleans Saints.
Gonzalez has been kicking with a groin injury that required an MRI on Monday, a source told ESPN's Adam Schefter.
"It's on me 100 percent," Gonzalez said after the game. "I can't be too mad at myself because I'm the one that did it, you know what I mean? I can't blame it on [anybody] else. It sucks because we were so close to that win, and it's been so long. I just let everybody down."
It's been 631 days since the Browns won a game. That came on Dec. 24, 2016, and although the Browns tied the Steelers on opening day this season, Gonzalez had what could have been a game-winning kick blocked in the final seconds.
ESPN's Pat McManamon contributed to this report.