The top-seeded Aztecs (25-6) meet Colorado State in the semifinals after the Rams ran over TCU 81-60.
New Mexico earned a semifinal berth against No. 20 UNLV by disposing of Air Force 55-39. UNLV finished the quarterfinal round by fighting off Wyoming 56-48.
Boise State (13-17) was down by eight points with less than 3 minutes to play, but began its rally with a 3-point shot from Anthony Drmic and a 3-point play by Derrick Marks that made the score 60-58 with 1:12 to go.
Franklin made it a four-point difference with a short shot but Drmic hit two free throws and, with 27 seconds left, Thomas Bropleh scored off a feed from Drmic for a 62-62 tie.
San Diego State called timeout and worked the ball around the outside with Franklin finally heaving a shot from just to the right of the top of the circle for the game-winner.
Franklin finished with 19 points, just short of the team-high 20 from Chase Tapley.
Drmic had 19 points for Boise State.
Colorado State had an unexpectedly easy time with TCU, which posted home wins this season over both San Diego State and UNLV. Pierce Hornung had 16 points for Colorado State (20-10) and Wes Eikmeier had 15.
Hank Thorns produced 19 points for the Horned Frogs (17-14).
New Mexico (25-6) built a 14-point halftime lead against Air Force (13-16) and was not threatened during the second period. Demetrius Walker made six 3-pointers and scored 19 points to lead the Lobos.
Michael Lyons had 17 points for Air Force, but made only 4-of-14 from the field.
UNLV (26-7) had only one player in double figures, with Mike Moser scoring 20 points, but the Rebels held Wyoming to 17 field goals in 58 attempts.
Wyoming (20-11) managed just 11 points in the first half and trailed by 22 early in the second period before getting as close as six in the final minute. Leonard Washington and Luke Martinez both scored 11 for the Cowboys.