Grant was drafted by the Redskins in the ninth round of the 1981 NFL Draft. The Redskins decided to turn him into a defensive lineman, even though he played offensive guard in college. After using his 1981 rookie year to adjust to the rigors of playing defensive tackle in the NFL, Grant joined the defensive line rotation starting in 1982.
Grant went on to become a fixture on Joe Gibbs' Redskins teams in the 1980s. At times paired alongside other Redskins defensive line standouts such as Dave Butz, Dexter Manley and Charles Mann, Grant was one of the contributors during a period (his Redskins career 1981–1990) that saw the Redskins make the playoffs six times, including three trips to the Super Bowl (two wins) and four conference championship appearances (three wins).
In the 1982 NFC Championship game against the Dallas Cowboys, one of the memorable moments from the 31-17 win was Grant's ten-yard interception return for a touchdown. He high-stepped his way into the end zone for the score. The crowd at RFK Stadium went into such a frenzy during and after Grant's play that many sections of the ballpark began swaying up and down, with some of them even compacting into each other from the force of the celebrations!
Grant went on to play in 141 games for the Redskins from 1981–90, finishing with 27 sacks and two interceptions. His best season was 1984 when he recorded a career-high eight sacks.