Washington quarterback Keith Price threw an interception on the first play of overtime that defensive lineman Kalafitoni Pole nearly returned for a game-ending touchdown, but he was caught at the 5.
The Cougars (3-9, 1-8 Pac-12) took possession and, after picking up a first down, Furney kicked a 27-yard field goal, leading to a crush of crimson and gray fans storming and celebrating at midfield.
Washington (7-5, 5-4) had a chance to win on the final play of regulation, but Travis Coons' 35-yard field goal attempt at the same end of the stadium was wide right after a bad snap.
In a rivalry known for unpredictability, the 2012 version will rank among the tops in Apple Cup lore. Washington led 28-10 after Bishop Sankey's second touchdown run late in the third quarter, but the Huskies wilted under a siege of self-inflicted mistakes with turnovers and penalties coming back in favor of the Cougars.
Carl Winston ran for three touchdowns for Washington State, matching the Cougars entire season total of touchdowns rushing entering Friday's game. His third TD with 7:26 left, combined with Jeff Tuel's 2-point conversion pass to Brett Bartolone pulled the Cougars within 28-25.
Washington was then forced to punt with 5:25 left after Price was forced to throw it away on third down. Tuel hit Gino Simone for 18 yards, his first catch of the game, to get into Washington territory and later a 9-yard reception to get inside the Huskies 30. The Cougars stalled after missing on third-and-1, but Furney connected on a 45-yard field goal to pull the Cougars even at 28-28 with 1:59 left.
Washington started at its 25 with all three timeouts. Price scrambled for a 7-yard run, and the Huskies were aided by a 15-yard personal foul call against Casey Locker moving the ball to the 47. Price hit Austin Seferian-Jenkins for 7 yards into Cougars' territory and found DiAndre Campbell for consecutive 11-yard throws to get to the Washington State 24 with 59 seconds left.
Sankey then carried twice to the 15, forcing Washington State to use its final timeout with 47 seconds left and the Huskies facing a third-and-1. Washington's school record-tying 18th penalty backed them up, and Sankey came up short on third down.
Washington coach Steve Sarkisian ran the clock down to 5 seconds before calling timeout. Coons was 3 of 4 inside 40 yards this year, but the snap back to holder Cody Bruns was low and the timing was off.
Washington State elected to go on defense first in overtime. Price was pressured by Logan Mayes on the first play of the extra session and he tried to throw for Sankey, but the pass landed in the arms of Pole. The big defensive lineman tried to end the game with the interception return, but was caught at the 5 by Bruns.
Because Pole didn't score, Washington State took possession at the 25 and on third-and-4, Tuel found Bartolone for 10 yards to the 9 and Mike Leach decided to end it there. Furney lined up and the kick was true, giving the Cougars their first win over their rivals since 2008.
Tuel finished 33 of 53 for 350 yards and two interceptions. The Cougars made plenty of mistakes, especially during Washington's 21-point third quarter, but showed resolve in the fourth quarter that was absent for most of Leach's miserable first season as coach at Washington State.
Price was 20 of 34 for 194 yards and two touchdowns, but also committed two costly turnovers. He had a fumble in the fourth quarter that led to Winston's third TD run. Sankey rushed 84 yards, seeing his streak of 100-yard rushing games snapped at three, but had two touchdowns bringing his season total to 15.
Washington saw its four-game win streak snapped and failed in its shot at its first 8-win season since 2001.