Mulkey was hospitalized earlier in the day after suffering a reaction to medication she was taking following removal of a kidney stone on Friday in Waco. Jones scored eight of her 21 points in overtime for Baylor (28-5).
Leon Barmore, whom Mulkey played for and coached with at Louisiana Tech, coached the Bears in the win.
Amber Gregg scored 23 points for UTSA (24-9), which was playing in its second NCAA tournament.
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The Roadrunners were bidding to become the first 15-seed ever to win a game in the NCAA tournament. The No. 2 seeds had won every matchup by at least five points since the field expanded to 64 teams in 1994.
Jordan Stark, who along with Southland Conference Player of the Year Monica Gibbs finished with 16 points, pushed the game into overtime when she hit two free throws with about 17 seconds remaining in regulation.
The Roadrunners led only briefly -- 71-69 -- in the extra period.
Jones hit back-to-back baskets in less than a minute to give the Bears a 77-72 lead before Stark was fouled on a 3-point attempt and hit all three free throws.
The Roadrunners used a 15-3 run to open the second half and go up 46-39 with more than 16 minutes remaining. Baylor continued to play from behind until Kelli Griffin hit a bucket with 9:24 left to put the Bears up 55-54.
Baylor seemed flat for much of the game and couldn't find its offensive rhythm. At one point, Morrow held her arms out with palms up, appearing not to know what she and her teammates were trying to accomplish on offense.
Baylor led 38-31 at halftime.
It was the Roadrunners second straight loss in an NCAA first-round game and the second straight against the Big 12 tournament winner.
Last year, UTSA lost to Texas A&M after the Aggies won the conference tournament.