Cleveland got Droughns in a trade for defensive linemen Ebenezer Ekuban and Michael Myers. The deal was completed Wednesday.
"I am extremely excited for this new opportunity with the Cleveland Browns," Droughns said on the Browns' Web site.
The 5-foot-11, 220-pound Droughns, a five-year NFL player, also caught 32 passes, two for touchdowns, last season.
Ekuban was the 20th overall pick in the 1999 NFL Draft by Dallas. Last season, he led the Browns with a eight sacks and recorded 39 tackles and one forced fumble.
Myers, who was a fourth-round pick by the Cowboys in 1998, had an off-year in 2004, making one sack and 34 tackles.