The Bears (11-1) used an 18-2 run that spanned both halves to take control and held on as the Gamecocks (8-5) cut a 17-point lead to 72-63 on Devan Downey's 3-pointer with 3 minutes left.
Udoh also had 12 points for his seventh double-double this season.
Baylor hadn't won seven in a row since starting the 2000-01 season 12-0 under disgraced former coach Dave Bliss.
It was South Carolina's first game since coach Darrin Horn dismissed one-time starting forward Mike Holmes for team violations.
Gamecocks leading scorer Devan Downey had 20 points on 6 of 25 shooting.
Nike Men's Baylor Bears Green Basketball Practice ELITE Stripe ShortsShop
Sebastien Bellin's friend Chris Young joins Ryan Smith on Outside the Lines to discuss his friend's bravery, courage and strength after surviving the Brussels bombings in March.
Andrew White III is leaving the Nebraska men's basketball team and will play elsewhere for his final college season.
A cameraman fell down trying to film Ben Simmons at the NBA Draft. Flashing back to his LSU days proves this is nothing new.