Plummer had 23 points in the first half as the Fighting Illini (20-8, 13-5 Big Ten) led by eight. They were ahead by 15 midway through the second half before the Wolverines (15-12, 9-8) rallied to give themselves a shot at a comeback win.
Caleb Houstan's 3-pointer with 2:03 pulled Michigan within two points, and Cockburn quickly responded with a jump hook.
Trent Frazier made a 3-pointer with 45.2 seconds left -- just before the shot clock expired -- for a seven-point lead. The win pulled Illinois within a game of Big Ten-leading and 13th-ranked Wisconsin.
Frazier finished with 17 points and Andre Curbelo scored 12 for the Illini.
Michigan was without suspended coach Juwan Howard, who missed the second of five games for hitting a Wisconsin assistant coach a week ago. Phil Martelli again filled in for Howard.
DeVante' Jones had a season-high 25 points and matched a career high with 10 assists for the Wolverines. Houstan scored 21 points, matching the season high he set in his previous game.
Hunter Dickinson had 13 points and 11 rebounds, playing in a matchup that he missed last month due to the effects of COVID-19 as Illinois won 68-53 at home.
Michigan's Moussa Diabate and Terrance Williams returned from one-game suspensions, stemming from the melee at Wisconsin. Diabate started and finished with 12 points. Williams did not play in the first half and had two points in four minutes of the second half.
Illinois: After losing three of the previous five games, the Illini put together a strong performance on the road to stay in the Big Ten championship race.
Michigan: The Wolverines have work to do to earn an NCAA Tournament at-large bid with a grueling schedule to end the regular season.
The Illini may stay in the same spot in the Top 25 after losing to No. 22 Ohio State by three points and beating Michigan.
Illinois: Finishes the regular season at home against Penn State on Thursday and No. 25 Iowa on Saturday.
Michigan: Closes the regular season, hosting rival Michigan State on Tuesday, the 25th-ranked Hawkeyes on Thursday and playing at No. 22 Ohio State on Sunday.