Jeremy Senglin added 18 points for the Wildcats (2-2), who advance to face Drake in the semifinal round on Tuesday night.
Weber State used an 8-0 run to lead 60-57 and Senglin's free throw with 1:16 left gave the Wildcats the lead for good at 61-60.
The Chippewas (2-1) had their chances late despite playing without last season's leading scorer Chris Fowler (16.2 ppg). The senior missed the start of the season due to an undisclosed injury.
Rayshawn Simmons turned it over with 15 seconds left and missed a layup with 2 seconds left that sealed it for Weber State.
John Simons had 17 points to lead Central Michigan, which will face Western Kentucky in the consolation bracket on Tuesday.