Collier's free throws came after Isaac Williams' jumper put MVSU up 84-83 with 19 seconds left. MVSU had one last possession, but a turnover and foul by Billy Jackson sealed it.
MVSU had the last shot in regulation and the first two overtimes, but missed each time in its upset bit.
MVSU's Vacha Vaughn forced overtime with his lone basket of the game, a layup with 50 seconds left.
Texas Southern's Malcolm Riley prompted the third overtime with a jumper with 25 seconds left.
Collier was 8 of 14 from the floor and made five 3-pointers for Texas Southern (9-11, 6-1 Southwestern). Madarious Gibbs added 17 points.
MVSU (2-18, 1-6) was led by Williams with 26 points. DeAngelo Priar had 17 points and 13 boards.