Furney's boot with 3:03 left lifted the Cougars (1-1, 1-0 PAC-12) over USC for the first time since 2000.
Damante Horton's 70-yard interception return for a score supplied Washington State with its only other points.
Connor Halliday went 26-of-38 for 215 passing yards with two interceptions and Kristoff Williams picked up 56 yards on six grabs for the Cougars.
Cody Kessler started at quarterback for the Trojans (1-1, 0-1) but was pulled following the first half after completing just 8-for-13 passes for 41 yards and an interception.
Max Wittek went 3-for-8 and was picked off on USC's final possession.