Rice also had seven rebounds and three assists and Illinois (9-3) had a tougher time than expected against Missouri (5-6), which is rebuilding under new coach Kim Anderson. Ahmad Starks added 13 points and Malcolm Hil had 10.
After Wesley Clark's jumper from the key tied it with 17.8 seconds remaining, Illinois left the game in Rice's hands. He dribbled for several seconds before stepping up the key and drilling a fade-away shot over defenders Keith Shamburger and Johnathan Williams III.
Williams had 15 points and eight rebounds for Missouri. Clark added 11 points, five rebounds and three assists and also tied it at 55 with 4:40 to go.
Illinois was unranked for the pre-Christmas rivalry game for the second straight season after losing three of five, although two of the setbacks were to No. 15 Miami and No. 7 Villanova.
It was just the eighth time in 34 games in the series with neither team ranked.
Jakeenan Gant scored off an inbounds play with 3:34 left to give Missouri its last lead at 57-55. The Tigers didn't score again until Clark tied it in the final seconds.
Illinois missed its first seven 3-point attempts but made two of its last three late in the half, with Starks connecting with 4.7 seconds left for a 32-30 lead.