The Aztecs trailed by 13 with 15 minutes remaining, but rallied to take its first lead since the opening moments of the game when Xavier Thames made a free throw with 2:48 left.
Boise State, however, regained a three-point advantage on a Derrick Marks jump shot with 1:32 to play.
The Broncos did not score again, suffering a turnover along with three missed shots by Marks the rest of the way.
Thames brought San Diego State within one by making a pair of free throws with 52 seconds left to go and Polee then came through with his game-winning shot. Marks tried and failed to hit a 3-pointer at the buzzer.
Thames went 10-of-11 from the foul line and scored 23 points to lead the Aztecs (20-1, 9-0 Mountain West Conference). The Aztecs lost to Arizona in the second game of the season and have won 19 in a row since.
Josh Davis was the only other San Diego State player in double figures with 10 points.
Mikey Thompson had 16 points for Boise State (15-8, 5-5 MWC) and both Marks and Thomas Bropleh had 15.