A 6-7 sophomore, Carney averaged 12.7 points this season. He surpassed that with 13 points in the first 14 minutes as the Tigers raced to a 27-15 lead and costed the rest of the way.
"It's great to have a game like this in the NCAA Tournament," said Carney, who was 10 of 17 from the floor and also grabbed 10 rebounds.
Memphis (22-7) earned its first NCAA Tournament win since 1995 and will meet second-seeded Oklahoma State in the second round.
South Carolina (23-11) was eliminated in the first round for the fifth straight time dating to 1974.
The Gamecocks made just two three-pointers in 11 attempts compared with nine from beyond the arc for Memphis.
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