Andrea Riley had 28 points and led Oklahoma State's run, scoring 10 straight points to give the Cowgirls a 54-53 lead with just over 3 minutes remaining.
Danielle Gant put Texas A&M up 58-54 with a three-point play and a breakaway layup, but Riley answered with a long 3-pointer with 25 seconds left, nearly falling into her own bench as the ball went through.
But that was it for the Cowgirls.
Franklin hit two free throws and Riley launched another 3-pointer with 12.3 seconds left, but the ball got wedged between the rim and the backboard. Franklin then sealed it with two free throws with 10 seconds left.
Gant finished with 11 points and Morenike Atunrase had 14 points on 4-for-8 shooting from 3-point range.
Riley had five steals, but was just 2-for-10 from 3-point range and had six turnovers. Oklahoma State (25-7) was 3-for-14 from long range and had just four players score, with Danielle Green getting 17 points, Shaunte Smith 12 and Kristin Hernandez two.
The women's Big 12 tournament is usually a showcase for powerful Oklahoma and Baylor teams, which had combined to win the past four championships -- three by the Sooners.
This year was different.
Missouri knocked off Oklahoma in the opening round on Wednesday and Baylor was gone a day later, losing by 16 to Texas in the quarterfinals, meaning the title game would be without either team for the first time since 2003.
Despite having good regular seasons, Texas A&M and Oklahoma State still seemed unlikely candidates to take their places.
The Aggies were 2-14 in the Big 12 just four years ago and hadn't won a conference title since upsetting Texas Tech in the 1996 Southwest Conference championship game.
Oklahoma State is two years removed from an 0-16 Big 12 season and had lost five straight in the Big 12 tournament. Neither team had reached the Big 12 championship game.