Utah posted three touchdown drives of less than 42 seconds and added two other scores on kick returns.
Clay returned a punt 46 yards for his first touchdown. In the second half Clay broke away for a 100-yard kick-off return.
The game marked the return of Utah quarterback Trevor Wilson, who was discovered to have a damaged intracranial artery while recovering from a concussion last season. Wilson missed the final three games but was cleared to play.
Wilson looked sharp against the overmatched Bengals and had 265 yards passing, while playing the first half. Dres Anderson caught four of Wilson's tosses for 111 yards.
Booker, a transfer from Washington State by way of Fresno State and American River college, had 78 yards on 10 carries.
After his own punt return gave the Utes a short field, Bubba Poole scored on a 6-yard run with 9:40 left in the first quarter.
ISU's Justin Arias scrambled away from the Utah rush and lofted a 3-yard touchdown to Broc Malcolm at the back of the end zone to tie the game at 7-7
Utah struck back when Booker took a screen pass 61 yards to set up Wilson's Superman-like leap for a touchdown with 2:59 left in the opening period. Wilson's few scrambles didn't show a hint of hesitation.
The Bengals worked their way right back down the field but Eric Rowe blocked an attempted 30-yard field goal and the Utes began to dominate. Four plays later, Booker scored on a 38-yard run.
Kenneth Scott, who missed almost all but a few plays last year with an injury, caught a 26-yard pass with 19 second left to make it 35-7 at halftime for Utah.
In the second half, Oklahoma transfer Kendal Thompson, who nearly won the starting job in fall camp, led the Utes on a scoring drive that culminated on Booker's second touchdown. Third-string QB Brandon Cox ran 16 yards for a fourth-quarter touchdown.
The Utes rolled to 589 yards of total offense and dominated despite not gaining a single turnover, though they did force 11 punts.
Arias threw for 158 yards on 21-for-40 passing and two touchdowns. ISU's Xavier Finney rushed 23 times for 133 yards.
Idaho State has now lost all seven matchups against Utah. The Bengals haven't won a road game since 2006 and last defeated a FBS team when they squeaked by Utah State 27-24 in 2000.