Delaney was selected in the second round of the 1981 NFL Draft by the Kansas City Chiefs. United Press International named him Rookie of the Year of the American Football Conference for the 1981 NFL season after he rushed for 1,121 yards, set four club records, and averaged 80.9 yards a game. Delaney's effort helped propel the Chiefs to a 9–7 record, the team's first winning season since 1973. In his rookie season, he was selected to the Pro Bowl after setting Chiefs records for most yards in a season (1,121), most yards in a game (193 vs. Houston), most consecutive 100-yard-plus games (three) and most 100-yard games in a season (five).