Moseley grew up in Livingston, Texas and played football at Livingston High School. After high school, he attended Texas A&M University from 1965 to 1966 and, later, Stephen F. Austin State University (SFA) from 1967 to 1969. He played quarterback while at both schools until his senior season at SFA when he made the switch to placekicker. In that season, he set Lone Star Conference records for most points in a game and most field goals in a season.

Moseley was selected 346th overall in the 14th round of the 1970 NFL Draft by the Philadelphia Eagles. He was dropped by the Eagles in 1971 and signed by the Houston Oilers only to be released again in 1972. He spent two years out of the NFL and moved back to Livingston, Texas where he installed septic systems. During this period, he sent letters to two dozen NFL teams and routinely practiced kicking with his wife. In 1974, the Washington Redskins signed him as a free agent.

In the strike-shortened 1982 season, Moseley made 20 of 21 field goals, a then-record 95.2 success rate, and was responsible for 76 points, and he became the only placekicker to ever win the Associated Press NFL Most Valuable Player. In the Washington Redskins' 27–17 victory in Super Bowl XVII over the Miami Dolphins, Moseley kicked two field goals and was successful on all three of his extra point attempts. During the following season, he led the NFL in scoring with 161 points.

In 1986, the popular 38-year-old Moseley was released by the Washington Redskins mid-season. He remains their all-time leading scorer with 1,207 points. He signed with the Cleveland Browns and retired at the end of the season, helping them win their divisional playoff game against the New York Jets with a game-winning field goal in double overtime despite missing two field goals in regulation and another in the first overtime period.