Jada Boyd added 18 points and 10 rebounds for the Wolfpack (22-2), who reached their third straight regional semifinal and 14th overall. They will face fourth-seeded Indiana or 12th-seeded Belmont.
USF (19-4) led by a point early in the third quarter before N.C. State used two big runs to pull away and take a 12-point lead into the fourth.
The Bulls pulled within eight points with 3 1/2 minutes to go. But the Wolfpack scored the next five points, capped by a 3 from Brown-Turner that made it 73-60 with 90 seconds to go.
The game was played at the Alamodome in front of a handful of people, including NCAA President Mark Emmert, who has been on the defensive since players took to social media to point out inequities in the NCAA's treatment of women's teams.
Elena Tsineke had 22 points to lead the Bulls, who were in the tournament for the seventh time. They made six 3-pointers in the first half to spark hope of an upset but hit just three more after halftime.
N.C. State overcame the absence of Kayla Jones, who injured her left knee on Sunday, and an off day by star center Elissa Cunane, who went 3 for 15 from the field. Cunane was 6 of 7 at the free-throw line and finished with 12 points and 11 rebounds as all five Wolfpack starters scored in double figures.
The Wolfpack used a 10-2 spurt, with four points from Boyd, to take a 47-40 lead midway through the third quarter. USF got within two before N.C. State erupted for a 12-0 run that made it 59-45 late in the quarter.
The Wolfpack took advantage of three turnovers in that stretch and USF coach Jose Fernandez received a technical foul to help them add to the lead.