Tapley's game-winning shot came after Derrick Marks had missed a free throw that would have given Boise State a three-point lead with 11 seconds to play.
The 3-pointer by Tapley was the only one he made in five attempts and he finished with 12 points. James Rahon led the Aztecs with 16 points and Jamaal Franklin added 14.
San Diego State is tied for third place in the Mountain West Conference race, 1 1/2 games behind front-running New Mexico. The Aztecs advanced to 17-5 for the season and to 5-3 in conference play.
Boise State (15-7, 3-5) was led by 16 points from both Marks and Mikey Thompson.