The Illini had a chance to win on Tracy Abrams' shot, but it bounced off the front of the rim as the final buzzer sounded.
Michigan (24-7), the regular-season champion, advances to Saturday's semifinals with its seventh straight win. It will face either fourth-seeded Nebraska or fifth-seeded Ohio State -- two teams it has not lost to this season. The Buckeyes and Cornhuskers play later Friday.
Conference player of the year Nik Stauskas scored 19 points to lead Michigan, which won despite scoring six points in the final 7½ minutes.
Illinois (19-14) was led by Abrams and Rayvonte Rice with 11 points each.
After leading most of the game, Michigan gave away the lead with 2:30 to go. But Morgan bailed out the Wolverines with his post move to win it.
The Big Ten's conference tournament is usually filled with frustrations for Michigan.
From vacating their only tourney title, in 1998, because of NCAA infractions to the 40-foot buzzer-beating heave Evan Turner made to give Ohio State a win, little has gone right for Michigan at this event. And it nearly happened again Friday after the Wolverines took a 13-point lead in the second half.
That's when Illinois' zone defense turned the game.
Following a media timeout with 11:26 to go, Stauskas made two free throws to give the Wolverines a 55-44 lead.
But the Illini, who came into the tourney with five wins in six games, scored 10 straight to cut the lead to one with 7:58 left.
Twenty-eight seconds later, Glenn Robinson III knocked down a 3-pointer to give Michigan a 58-54 lead. Robinson finished with 15 points and seven rebounds.
Illinois then resumed its charge, finally taking a 59-58 lead on Rice's driving layup with 4:53 to go.
Michigan tied the score at 61 on Derrick Walton's 3 with 3:10 remaining, but all the Wolverines could muster the rest of the way was one free throw from Stauskas and Morgan's layup with 7.9 seconds to go.
After two timeouts, Illinois gave the ball to Abrams who drove into the lane expecting contact. But the shot bounced cleanly off the front of the rim and no foul was called.