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.
Miami has lost running back Gus Edwards for the entire season because of a left foot injury.
Virginia Tech will wear a decal on its helmets against No. 1 Ohio State on Monday night to honor slain Roanoke television journalists Alison Parker and Adam Ward.
South Carolina football players will wear stickers on their helmets this season to honor the nine people who died in a shooting at Emanuel AME Church in Charleston this past June.
Washington has found its starting quarterback, but coach Chris Petersen said the Huskies will keep that information under wraps until Friday's season opener at No. 23 Boise State.
Joey Counts scored on a 1-yard run on fourth-and-goal with 2 seconds left to give Montana a 38-35 victory over four-time defending FCS champion North Dakota State on Saturday.
Transfer Vernon Adams will be the starting quarterback when Oregon begins its season against Eastern Washington on Sept. 5.