Chicago (10-7 overall, 3-3 UAA) trailed 77-73 with 22 seconds remaining in the extra period. Woodhead hit a layup with 15 seconds left to cut the gap to 77-75. Carnegie Mellon (14-3, 4-2) missed two foul shots four seconds later, setting up Woodhead's winning three-pointer.
The Tartans jumped to an early 19-9 lead, but Chicago rallied to take a 28-26 lead at the intermission.
Chicago held the advantage for the first 9:00 of the second stanza before the Tartans tied the game at 42-42 on a basket by Nate Maurer.
The final 11:00 of regulation featured eight ties and five lead changes. A Geoff Kozak three-pointer put the Tartans ahead 65-64 with 34 seconds to go, before Clay Carmody connected on one of two free throw attempts with 17 seconds left to force the overtime.
Jesse Meyer led the Maroons with 18 points and six rebounds, while Woodhead finished with 16 points. Matt Corning and Carmody contributed 13 and 10 points, respectively.
Maurer led all scorers with 23 points for the Tartans. Clayton Barlow-Wilcox added 16 points and a game-high eight rebounds, and Bary Dunn chipped in 10 points off the bench.