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.
Ryan Bass ran 19 times for 121 yards for Idaho, which didn't lead until Blackman found Najee Lovett for a 19-yard score to make it 27-20 with 4:51 left in the third quarter.