The win, which fell on coach Bruce Pearl's 50th birthday, sends Tennessee (26-8) against No. 14 seed Ohio University (22-14) Saturday at 3:30 in the round of 32. On a day when underdogs prevailed in seven of 16 first-round games, the Bobcats posted the biggest shocker of all - a stunning 97-83 thumping of third-seeded Georgetown.
A San Diego native who had made just 28.9 percent of his 3-point tries coming in, Goins sank 4 of 5 shots from beyond the arc on this night and finished with a career-high 15 points.
"I just let the game come to me," he said. "I took my open shots and knocked 'em down. I was feeling pretty good."
San Diego State rallied from a 34-26 halftime deficit to grab a 45-44 lead with 9:20 left. Tennessee eased ahead 55-50 with 3:32 to play but committed turnovers on four consecutive possessions, enabling the Aztecs to pull within 55-54. Brian Williams finally stopped the bleeding by hitting a couple of free throws with 1:03 left to provide a 57-54 lead.
D.J. Gay answered with a pair of foul shots that narrowed the gap to 57-56 with 51.7 seconds left. Tennessee's J.P. Prince missed two foul shots with 46.9 seconds left but Wayne Chism rebounded the second miss and got the ball to Goins, who drained a 3-pointer that expanded the lead to 60-56 with 13.8 seconds left.
"I felt confident," Goins said. "Even before they kicked the ball to me, Coach (Pearl) was yelling, 'Stick, stick, stick.' They kicked it, I shot it, and it went in."
Due to questionable decision-making Goins generally watches the final minutes of Vol games. But Pearl showed faith in the 5-10 backup point guard by playing him down the stretch this time.
"My assistant coaches suggested late in the game that I put Melvin in with Bobby (Maze)," Pearl said. "I did it for defensive purposes and I did it because Melvin was playing well."
Goins may have been inserted for defensive purposes, but he wound up making the most important offensive play of the night. Pearl wasn't surprised when the 3-pointer dropped but he was relieved.
"When that ball tapped out to Melvin I yelled 'Stick,' and Melvin took the shot," Pearl recalled. "I don't know if that was a smart play or the right play but we had struggled so much offensively - and Melvin had knocked three down before - so I wanted to give him the confidence to take that shot."
Moments after hitting the biggest basket of the game, however, Goins made the biggest mistake of the game. With 9.5 seconds left he fouled Gay as he unloaded a desperation 3-point attempt. Gay made all three free throws, trimming the lead to 60-59.
Chism sank a pair of clutch free throws with 7.4 seconds left, however, and the Vols held their breath as San Diego State's Kawhi Leonard missed a potential game-tying 3-pointer at the buzzer.
Like Goins, Prince scored 15 points for the Vols, who posted a first-round NCAA Tournament victory for the fourth time in Pearl's five years at the helm. Chism and Maze chipped in 11 points each.
Gay scored 16 to pace San Diego State, which finishes 25-9. Leonard added 12 points and 10 rebounds.