The Longhorns had lost two games by a combined total of nine points, with the two losses costing them a Southwest Conference title.
Husky defensive lineman Stafford Mays recovered quarterback Rick McIvor's fumble late in the first quarter in Longhorn territory.
On the second play of the Longhorns' ensuing drive, strong safety Greg Grimes recovered McIvor's fumble at the 25-yard line, giving Washington another shot at scoring.
But on Washington's next possession, Flick fumbled the ball to defensive lineman Kenneth Sims, who recovered it in Texas territory.
In gusty winds, the rest of the game was a defensive affair, with the Longhorns' fumbles in the first half proving costly.