The Utes (6-3, 3-2 Mountain West Conference) scored on four offensive possessions in the first half.
Johnson marched Utah down the field on the game's opening possession, connecting with running back Darryl Poston on a 15-yard touchdown pass to cap an 11-play, 80-yard drive.
The Rams (1-7, 1-4) countered with a field goal on the following drive after being stopped on three consecutive plays inside the Utah 10.
Utah held them out of field goal range the rest of the game and finished with six sacks.
The Utes scored on their next three possessions, including another touchdown pass by Johnson. This time the junior quarterback found wide receiver Brian Hernandez open near the sideline for the 4-yard score.
Colorado State was doomed by its poor third-down conversion rate. The Rams were 2-for-13 in that category, while the Utes were 6-for-12.
Rams quarterback Caleb Hanie was replaced at the beginning of third quarter by Billy Farris. He returned in the fourth quarter, but was ineffective as he threw for 94 yards and an interception. His 13 interceptions are second most in the conference.
Darrell Mack ran for 151 yards for the Utes.
Stowers' 1-yard touchdown came late in the game.