The Browns sent offensive tackle Tyrone Wheatley Jr. to the Patriots for Strong, a fourth-round pick last season by New England.
Cleveland made the move two days before NFL teams are required to trim their rosters to 53.
The 5-foot-11, 205-pound Strong, who played at South Dakota State, appeared in 15 games as a rookie. He gained 100 yards on 10 carries and scored a touchdown, adding seven catches for 42 yards.
With second-year back Jerome Ford out with a hamstring injury, Cleveland has been looking to add another experienced runner to slot behind Chubb, a four-time Pro Bowler who for a career-high 1,525 yards last season.
Cleveland's running back room has been in flux since the team decided not to re-sign Kareem Hunt during the off-season and D'Ernest Johnson signed as a free agent with Jacksonville.
Wheatley wasn't guaranteed a roster spot with Cleveland. He's been supplanted on the depth chart by rookie Dawand Jones, who has had a solid training camp and preseason.
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Tom Withers, The Associated Press