Kawhi Leonard missed a 3-point attempt at the buzzer that would have forced overtime and the sixth-seeded Volunteers advanced to take on Ohio University in the second round.
Ohio stayed alive in the tournament by shocking Georgetown earlier in the evening 97-83.
Tennessee made just 37 percent of its shots and Chism hit only 4-of-12 from the field but he came through with the two critical foul shots at the end of the game.
San Diego State, which won the Mountain West Conference tourney to earn an NCAA bid, trailed by four points when D.J. Gay was fouled attempting a 3-pointer with 10 seconds left.
Gay made all three foul shots to cut the Tennessee lead to one. Chism then responded with his free throws after being fouled by Kelvin Davis. The Volunteers did a good job on defense slowing down the Aztecs as they tried to rush up the floor following the Chism foul shots and Leonard's attempt to tie it was not close.
J.P. Prince led Tennessee (26-8) with 15 points. Chism and Bobby Maze both had 11.
San Diego State (25-9) got 16 points from Gay. The Aztecs were good on just 3-of-18 3-point tries.
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