The Browns also announced they waived quarterback Joel Stave, who had been signed a week ago.
Vogel was the Packers' punter last year and averaged 44.4 yards per punt. His net average of 41.6 yards was a single-season franchise record. He had 19 punts inside the 20-yard line and just two touchbacks.
The Packers waived Vogel last week after they selected punter JK Scott in the fifth round of this year's draft.
Vogel will compete with Britton Colquitt for the punting job in Cleveland. Colquitt averaged 47.6 yards per punt last season.
Vogel was originally signed by Green Bay as an undrafted free agent in 2017.