"That was really good," Vol coach Bruce Pearl said on the post-game show. "Let's face it: We've had some last-second possessions where we've not gotten as good a look. Melvin took it into the lane and got a nice shot he can make. That's a game-winning shot by a point guard. It was good that we got it ... that Melvin got it."
Following a Carolina turnover, Goins made a free throw with 19.8 seconds left and Harris added another with 8.3 seconds remaining as the Vols went up 73-66. The Gamecocks' Bruce Ellington hit a 30-footer at the buzzer to secure the final score.
With the win, Tennessee improves to 18-12 overall and 8-7 in SEC play. The Vols finished 5-3 in conference road games.
"I wish we could play as well at home as we do on the road," Pearl quipped. "I also wish we could play South Carolina more often."
Pearl now stands 12-2 vs. the Gamecocks and has won the last nine meetings.
The loss drops South Carolina to 14-14 and 5-10.
With 6-10 senior center Brian Williams sidelined by back problems, Tennessee was outrebounded 35-27. Carolina parlayed 20 offensive rebounds into 22 second-chance points.
"If we don't give them 22 second-chance points on 20 offensive rebounds it's a very convincing win for Tennessee," Pearl said. "But we're tremendously undersized in there."
Tennessee made four fewer field goals (22) than the Gamecocks (26) but won the game at the free-throw line, cashing in on 25 of 34 attempts, compared to Carolina's 12 of 13.
Harris finished 7 of 9 from the field and 11 of 14 from the foul line en route to a career-high 25 points. Hopson hit 9 of 17 shots (3 of 7 from 3) and chipped in 24.
"Tobias Harris played like an All-American," Pearl said, noting that the 6-8 freshman produced clutch foul shots and key rebounds "and every one of them was a 'traffic' rebound."
Muldrow was brilliant in defeat, contributing 24 points, 12 rebounds and 5 blocks for the Gamecocks. Ramon Galloway added 15 points off the bench.
The Vols seemed ready to blow the game open in the first half, using a 19-6 explosion to turn an 11-9 lead into a 30-15 bulge. The lead peaked at 40-20 on a Hopson dunk with one minute left to halftime.
"We came out awfully sharp," Pearl said. "When you go on the road you've got to do that ... send a message that you're here to take care of your business."
Ramon Galloway hit a 3 to narrow the gap to 40-23 at the half, then scored seven consecutive Carolina points during a second-half spurt that saw the Cocks trim the deficit to 58-54 with 8:03 remaining.
The Vols went up 69-58 on a Hopson dunk with 2:43 to go but Galloway hit a 3-pointer and a pair of free throws, then Muldrow scored a three-point play that narrowed the gap to 69-66 with 1:06 remaining and set the stage for Goins' late heroics.
Tennessee shot a healthy 51.2 percent from the floor, including 36.4 percent (4 of 11) from 3.
"We handled that zone really well; I was surprised they stayed zone as long as they did," Pearl noted. "Even in the second half when we weren't as sharp offensively, we got to wind the shot clock down.... I was surprised we didn't see more aggressive man-to-man from them."