Washington graduated from Lincoln High School in Port Arthur, Texas where his father coached football. Washington had a stellar college football career at the University of Oklahoma where he was a two time First-team All-American and finished third in the Heisman Trophy balloting in 1974 and fifth in 1975. He finished his career at Oklahoma with 4,071 career rushing yards. He is a member of the College Football Hall of Fame.

Washington was drafted fourth overall in the first round of the 1976 NFL Draft by the San Diego Chargers. A knee injury forced him to miss the entire 1976 season for the Chargers. In 1977 he played sparingly, appearing in 13 games while rushing for 217 yards and having 244 yards receiving.

He was traded along with a 1979 fifth-round selection (131st overall–traded to Detroit Lions for Greg Landry) from the Chargers to the Colts for Lydell Mitchell on August 23, 1978. The transaction was the result of Mitchell’s acrimonious contract dispute with Colts management in which he accused team owner Robert Irsay of bad faith bargaining and racial discrimination.

In 1978, his first year with the Colts, he had 958 yards rushing, which was a career high.

His most memorable performance was on September 18, 1978 in Baltimore’s 34–27 victory over the New England Patriots on Monday Night Football, when he had a hand in three of the four touchdowns scored by the Colts in a fourth quarter in which both teams combined for 41 points. He helped lead off the scoring by throwing a 54-yard option pass to Roger Carr. Washington followed that up with a 23-yard touchdown catch from Bill Troup. Washington broke a 27–27 deadlock by scoring the game-winner on a 90-yard kickoff return in a driving rainstorm. It was scored in the final seconds of the game after the Patriots came back from 27 to 13.[1][2]

In 1979 he led the NFL with 82 receptions. He also had 750 yards receiving and 884 yards rushing along with seven touchdowns. This led Washington to being voted to the 1979 Pro Bowl.

On April 28, 1981, the Washington Redskins obtained Washington by trading a second-round pick in the 1981 NFL Draft to the Baltimore Colts.

During his first year with the Redskins in 1981, Washington combined with future Hall of Famer John Riggins to give the Redskins a formidable running game. Washington led the team in rushing with 916 yards and receptions with 70 to go along with 558 yards receiving. The strike shortened 1982 season was mostly a disappointing season for Washington as he only recorded 190 yards rushing during the 9 game regular season and only touched the ball seven times during the Redskins run through the playoffs where they defeated the Miami Dolphins in Super Bowl XVII.

Washington returned to his normal dual threat capabilities in 1983 as he recorded 772 yards rushing and 454 yards receiving as the Redskins returned to the Super Bowl only to lose Super Bowl XVIII to the Los Angeles Raiders.

During the 1984 season, Washington only played in 7 games for the Redskins and recorded 192 yards rushing.

He had worked as a financial adviser for Wells Fargo. With former basketball player Julius Erving, Washington fielded a NASCAR Busch Series team from 1998 to 2000. In May 2007, Washington returned to the University of Oklahoma, to serve as the special assistant to the director of athletics/executive director of the Varsity O Association.