Tyree Appleby scored on a short baseline jumper with 32 seconds left to give the Vikings (12-22) the lead.
Cleveland State then got a stop with 14.5 seconds left, Appleby missed the front end of a 1-and-1, and the Vikings got another stop as time expired to secure the win.
The Vikings, who upset top-seeded Northern Kentucky on Saturday, will face No. 2 seed Wright State in the championship on Tuesday.
Oakland (19-14) had the largest lead of the second half at 31-26 on Kendrick Nunn's layup with 12:47 to go. Carpenter's 3 cut the deficit to two and neither team led by more than a possession the rest of the way.
Nunn, the nation's second-leading scorer entering the game (26.1 per game), finished with 19 points on 7-of-24 shooting. Jalen Hayes added 20 - the pair accounting for 39 of the Grizzlies' 43 points.
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