Maurer scored on a layup with 1:15 left in the game to break a 73-73 deadlock. Chicago had a chance to tie on its final possession, but a jumper by Brandon Woodhead rolled off the rim as time expired.
Carnegie Mellon (17-5 overall, 9-3 UAA) led 42-39 at the conclusion of a first half which featured eight lead changes.
The Tartans built their largest lead of the second half at 64-53 with 8:16 left in the contest. Chicago rallied to tie the game at 66-66 on a three-point basket by Brian Cuttica at the 4:10 mark.
The teams then traded baskets until Maurer's decisive shot with 1:15 to go.
Clay Carmody led the Maroons with 19 points, while Cuttica finished with 14 points to go along with six assists.
Mike Dolezal and Justin Waldie each contributed 10 points. Dolezal also collected nine rebounds and three blocked shots as the Maroons closed out the home portion of their 2004-05 schedule.
Clayton Barlow-Wilcox and Michael Divens scored 12 and 11 points, respectively, for the visiting Tartans, who remained alive in the UAA title race entering the final week of the regular season.