Leggett's run broke a 3-3 tie with 11:57 remaining in the second quarter.
Larry Johnson ran for 92 yards and a touchdown, and Connor Barth kicked field goals of 38 and 27 yards for the Chiefs (2-10).
Kansas City quarterback Tyler Thigpen completed 15-of-22 passes for 162 yards.
JaMarcus Russell connected on just 10-of-28 passes for 132 yards for Raiders (3-9).
Justin Fargas scored on a 1-yard run and Sebastian Janikowski booted field goals of 25 and 51 yards for Oakland.