Ball State (1-0) will meet UCLA or Houston in the semifinals on Tuesday.
A 3-pointer by Ball State's Chris Williams tied the game at 91-91 with 45 seconds remaining, and Kansas (0-1) failed to score on its ensuing possession.
The Cardinals gave the ball to Jackson at the top of the key in a play designed to find a shot for Williams. But Williams was double-teamed, forcing Jackson to improvise. Jackson easily beat freshman Aaron Miles before scoring between two defenders to give the Cardinals the lead.
Kansas forward Drew Gooden missed a desperation shot as time expired, sending Ball State's bench into a frenzy.
Williams led Ball State with 24 points while Jackson chipped in 23. The Cardinals survived the loss of leading scorer Theron Smith, who had 19 points but fouled out with just over four minutes left.
Gooden made 12-of-21 shots for 31 points while Nick Collison finished with 15. Kirk Hinrich added 14 points and six assists for the Jayhawks, who saw their 36-game November winning streak come to an end.