Xavier Thames scored 18 points and Chase Tapley 17 for the fourth-seeded Aztecs (22-9), who advanced to Friday's semifinals against No. 1 seed New Mexico.
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Anthony Drmic had 20 points and Derrick Marks 14 for the fifth-seeded Broncos (21-10), who figure to be sweating out NCAA tournament selection day on Sunday.
The Aztecs led for much of the game but were unable to put a knockout punch on Boise State, who erased a 13-point second half deficit while San Diego State went scoreless for 3:39. Two free throws by Drmic gave the Broncos their only lead, 50-49, but Franklin quickly responded with a pair from the line and led the Aztecs' charge down the stretch.
Franklin's thunderous transition dunk made the score 66-59 with 43 seconds remaining, as San Diego State did what few teams in the nation do better -- close out a game. The Aztecs have won 91 consecutive contests it led with five minutes remaining in regulation.