The Utes had managed little offense all night until they put together an 80-yard drive after TCU's second missed field goal of the fourth quarter.
Johnson hit seven passes on the march, including a 23-yarder to Brent Casteel that put Utah in TCU's end of the field.
Utah, with a 10-0 record, remained one of six unbeaten major college teams, along with Ball State, Boise State, Alabama, Texas Tech and Penn State. The Utes, at 6-0 in the Mountain West Conference, have a one-game lead in the loss column over TCU, Air Force and BYU.
TCU fell to 9-2 and 6-1. Its only other loss this year has been to Oklahoma.
Not only did the Horned Frogs miss two field goals in the final period, they threw away two other chances at field goal attempts in the first half when quarterback Andy Dalton was sacked for big losses on third down. Dalton was also intercepted twice, including a pickoff by Robert Johnson with 11 seconds remaining at midfield that clinched the outcome.