The Spartans (10-21) overcame a 50-37 deficit with 9:30 left and went on a 15-2 run that tied the score at 52.
Hawkins scored seven straight points to again tie the game at 54 with 2:45 remaining.
Marion Thurmond hit a 12-footer for a 56-54 lead, but Omar Seli Mance's 3-pointer put the Owls (10-19) back ahead at 57-56 with 49 seconds remaining. That set up Hawkins' winning jumper with about 33 seconds left.
Hawkins led all scorers with 23 points and Thurmond added 10 for San Jose State.
Michael Harris paced Rice with 18 points and Mance had 14.
The ninth-seeded Spartans will meet No. 1 seed Hawaii (24-5) on Thursday.