Rayvonte Rice added 14 points for Illinois (9-2), which snapped a four-game losing streak in the 33rd renewal of the annual neutral-court matchup.
Illinois scrambled on its final possession but got a bit of a break when Abrams was fouled by Johnathan Williams III driving the lane. Abrams was 7 for 10 at the line.
Tony Criswell was long on a desperation 3-pointer from well beyond the line at the buzzer.
The point total was a season-low for Missouri, which scored 72 or more its first nine games but beat Western Michigan 66-60 in its previous game.
Illinois was unranked for the pre-Christmas game for the first time in four years, and perhaps learned from a loss at No. 15 Oregon its last time out.
The last four years, the schools entered with a combined 72-7 record.
Among those attending were Hall of Fame baseball manager Whitey Herzog, Missouri Gov. Jay Nixon and the football coaches from both schools, Missouri's Gary Pinkel and Illinois' Bob Beckman.
The schools traded runs in the second half that kept it tight, 10-1 by Missouri to go up by near the midway point and 9-0 by Illinois with a pair of three-point plays by Abrams for a two-point lead with 6:17 to go.
Illinois climbed out of an early hole behind 3-point shooting, hitting six of its first 10 -- by six players -- and led 31-27 at the half. Missouri led 8-0 and had a nine-point cushion at 15-6 after Rice's second foul with 13:47 to go in the half.
Illinois answered with a 10-0 run the next 31/2 minutes, including 3-pointers from Kendrick Nunn and Abrams, and scored seven in a row late in the half for a six-point lead ended by Clarkson's driving basket with five seconds to go.