Second-seeded Illinois State (23-8) faces either Southern Illinois or Northern Iowa in a semifinal contest Saturday.
Seventh-seeded Missouri State, which won five of its last seven games, ends the season 17-16.
The Redbirds outscored Missouri State 16-7 over the final 6:26. Emmanuel Holloway chipped in with 11 points. Brandon Sampay came off the bench to add 10 points and seven rebounds for Illinois State.
Deven Mitchell and Dale Lamberth led Missouri State with 18 points each.
The Redbirds have won five in a row and seven of their last eight. They held Missouri State scoreless during a stretch of 4:16 late in the second half.
Eldridge, a first-team all-conference selection, hit four free throws in a span of 27 seconds to put his team up to stay 59-51. Missouri State climbed to within 61-58 on a short jumper by Mitchell. But Richardson sank one of two free throws with 24 seconds left to halt the rally.
Missouri State took a 51-47 lead on a layup by Mitchell before Illinois State went on a 12-0 run to take control. Eldridge scored six points during the streak. Richardson and Alex Rubin added 3-point plays.
Illinois State is looking to reach the NCAA Tournament for the first time since the 1997-98 season.
- Jay (The Real MSU)