The Cougars (8-8, 1/3 Pac-12) came back from a 12-point deficit for their first Pac-12 win.
"It was imperative we got a win at home. Washington State coach Ken Bone said. "It was our only home game out of the last four, you better win it, because it doesn't get any easier from this point forward," Bone said.
Washington State lost in overtime on Wednesday against Colorado. The Cougars were determined to make sure they closed out against Utah, who also lost a close one, 59-57, to Washington.
"You could say this was Washington State's defense, and it was good, but we have lost a couple down the stretch by not executing, so it rests on us," Utah coach Larry Krystkowiak said.
With 56 seconds on the clock, Delon Wright stole the ball from Johnson and went the length of the court for a dunk to pull the Utes within one, 47-46.
After an offensive foul on Johnson, Utah's Brandon Taylor traveled and put the ball back in the hands of Washington State.
The Utes (12-4, 1-3) were forced to foul and Johnson made a pair of free throws. Washington State made a decision to foul Delon Wright on the next play, and it paid off -- Wright missed both free throws.
"Strategically I believe in fouling, if it is under or around six seconds, and it was the right situation and it worked out," Bone said.
Wright had one more look at a three but missed short as time expired.
Cougars shot only 30 percent from the field in the first half. Utah took a 12-point lead as the Cougars once again struggled to find offense early. Washington State also played without leading scorer DaVonte Lacy, who is recovering from a rib injury.
Dakarai Tucker hit a jumper to start the second half, but Utes went cold did not hit a field goal for six minutes. Washington State capitalized and tied the game at 31 with 10:56 left.
Ike Iroegbu nailed a long three, followed by a fast break layup to give Washington State a 36-35 lead -- their first since the opening minutes of the game.
Washington State's Dexter Kernich-Drew hit a long three at the top of the key, followed by a jumper to give the Cougars a 47-44 edge at 1:55.
Kernich-Drew scored nine-second half points and hit a late game three for Washington State (8-8, 1/3 Pac-12). D.J. Shelton had eight points and nine rebounds.
Jordan Loveridge scored 15 points and hit the only two 3-pointers for Utah (12-4, 1/3 Pac-12). Brandon Wright added 14 points for Utah, which shot a season-low 29.5 percent (13 of 44) from the field.