Harris, a freshman, was 11 of 14 from the foul line in surpassing his previous high of 21 set in his first Southeastern Conference game against Arkansas.
Hopson scored 16 of his points in the opening half as the Vols (18-12, 8-7) opened a 20-point lead.
The Gamecocks (14-14, 5-10) cut the margin to 69-66 with 1:06 to go. But Tennessee's Melvin Goins followed with a jumper and South Carolina could get no closer.
Senior Sam Muldrow had a career-high 24 points to lead the Gamecocks in his last regular-season game at Colonial Life Arena. Muldrow also collected 12 rebounds.
Tennessee, once a sure-fire NCAA tournament lock, came into this one dragging a bit after five losses in its past seven games.
And usually, the Gamecocks are just the cure for what's ailing the Vols. That remedy looked good again in this one, especially early on when Tennessee couldn't miss and South Carolina couldn't hit much of anything.
Bruce Ellington's 3-pointer tied the game at 7, and then Tennessee took off on a 20-6 run.
Hopson scored eight points in the surge. He took a pass from the right side of the foul line and glided unimpeded for a jam, then followed with a silky smooth 3-pointer. By the team the coach's son, Steven Pearl, put in a nifty reverse, the Vols were ahead 27-13.
Tennessee made about everything it threw at the bucket. It hit 10 of its first 13 shots and ended the half at over 61 percent (13 of 21). Hopson led the way with 16 points as the Vols ended the period up 40-23.
South Carolina was 4 of 15 at one stretch and had no answers to Hopson's high-flying game or Harris' ability underneath. The 6-foot-8 freshman more than made up for the absence of backup center Brian Williams, who remained in Knoxville with back problems.
The Gamecocks caught fire after the break as they tried to make Muldrow's senior night one to remember.
Ramon Galloway's basket ended a 31-18 run and put South Carolina back in it, down 58-54. However, Harris keyed an 11-4 Vols surge with two foul shots and a pair of buckets.
South Carolina had one final charge as Galloway hit a 3-pointer and two foul shots and Muldrow completed a three-point play to get within 69-66.
That was as close as the Gamecocks could get. Muldrow came out of the game to applause with 8 seconds left and the crowd broke into a "Thank you, Muldrow" chant.
Muldrow leaves as South Carolina's all-time blocks leader.
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