Idaho (0-4) had tied the game on the final play of regulation, when Dominique Blackman tossed an 8-yard TD pass to Mike Scott in the corner of the end zone. Trey Farquahar's extra point made it 34-all.
Farquahar's 49-yard field goal put the Vandals ahead in overtime, but Smith and Claiborne pulled it out for the Cowboys (1-3).
Smith was 23 of 35 for 365 yards and was not intercepted, and Chris McNeill caught eight passes for 219 yards. Blackman finished 34 of 46 for 306 yards and had three touchdowns and no interceptions.
GameDay 100 Leonard Fournette trucks over opponent to score 22-yard rushing touchdown.
Joel Honigford is Michigan's fourth ESPN 300 commitment this month and could develop into a quality starter.
ESPN.com's Ivan Maisel calls into The Paul Finebaum Show to make early predictions for this season's College Football Playoff.
Hugh Kellenberger joins The Paul Finebaum Show to discuss Mississippi State's 2016 season.
Hugh Kellenberger calls into The Paul Finebaum Show to preview the Rebels' season opener against the Seminoles.
The Missouri Tigers will hope a still-stout defense can hold the fort will the offense finds itself before a rough October stretch.