Antonio Barton and his brother Will Barton combined for 46 points and 16 rebounds but couldn't keep Tennessee from clawing back from a 16-point deficit and force the extra periods.
Tennessee's Jeronne Maymon finished with 32 points and a tournament game record 20 rebounds.
Memphis led 40-24 in the first half but a Maymon shot with 7 seconds to play that tied the score at 83-83 and led to the first overtime. Chris Crawford's layup for Memphis with 29 seconds to go made the score 91-91 at the end of the first extra period.
Maymon had four points in the second overtime but Tennessee scored just two more points while Memphis used an Antonio Barton shot with 1:16 to go to break the final tie.
Will Barton ended with 25 points and 10 rebounds, and Antonio Barton added 21 points and six rebounds for Memphis (2-1). Adonis Thomas had 19 points for the Tigers.
Maymon, who made 16-of-17 free throws, led all scorers and Trae Golden made 14 points for the Volunteers (2-2).