The Illini (19-13) used runs of 9-0 in the first half and 11-0 in the second half to improve to 6-2 all-time against Indiana in the Big Ten Tournament.
The victory sends Illinois into a second-round matchup on Friday with Big Ten regular season champion No. 8 Michigan.
Abrams stymied a late rally by the Hoosiers (17-15) with a 3-pointer and two free throws that turned a late 53-52 Illinois lead into a 58-52 advantage.
Rayvonte Rice added 13 points and Kendrick Nunn scored 10 for the Illini.
Yogi Ferrell led the Hoosiers with 14 points. Will Sheehey added 13 points -- 11 of them in the first half -- and Troy Williams scored 11 for Indiana.
Illinois opened the game with a 9-0 run, taking advantage of Indiana's cold shooting and carelessness with the ball. The Hoosiers missed their first nine shots and had nine turnovers in the first half.
But Indiana answered with a 14-2 run, capped by a Ferrell 3-pointer, to take its first lead.
It didn't last long, with Noah Vonleh being called for goaltending on a basket by Rice which put the Illini back in front 15-14.
Illinois led 30-28 at halftime. Indiana was 6-of-10 on 3-pointers in the first half but just 2-of-13 inside the arc.
Troy Williams tied the game at 30 early in the second half on a dunk, then the Hoosiers took a 32-30 lead on two free throws by Stanford Robinson. Robinson, who was 0-for-2 from the line in the first half, was 4-of-4 in the second half.
The Hoosiers built a 38-33 lead to open the second half, but the Illini rallied to tie the game. Abrams missed two free throws that would have given Illinois the lead. But a second goaltending call on Vonleh, again on a Rice shot, put Illinois in front 40-38.
The Illini would stretch the lead to 44-38, prompting an Indiana timeout. Vonleh got free inside for a dunk, but the Hoosiers left Jon Ekey open for a 3-pointer that put the Illini in front 47-40.
Ferrell hit two free throws, Sheehey hit a jumper and Ferrell scored on a drive that pulled Indiana within 47-46.
Abrams answered with a 3 to make it 50-46 Illinois. Williams scored on a putback to make it 50-48, but Indiana could get no closer than 53-52.
Abrams then scored the next five points to secure the victory.
Illinois improved to 5-1 in first-round games in the Big Ten Tournament and won as the No. 9 seed for the first time. The Hoosiers fell to 10-17 in the conference tournament and fell to 1-2 as the No. 8 seed.