Chicago State led 41-37 after Quinton Pippen's jumper 4 minutes into the second half, but an 18-4 run over the next 7 minutes, capped by two Provost free throws, put the Broncs ahead 55-45.
The Cougars pulled within 66-65 on Corey Gray's 3-pointer with 14 seconds left, but Provost made two more free throws before Gray missed a 3-pointer at the buzzer.
Chicago State still leads 7-5 in the all-time series between the schools, but Texas-Pan American has won the last three meetings.
Guard Kobe Paras, who was part of the freshman class, does not meet academic conditions and won't be attending UCLA.
Shaquille O'Neal said he wants his son, Shareef, to play for coaches who can help him reach the NBA and develop more discipline.
College basketball coaches across the country will hit the recruiting trail hard in July to target the Class of 2017's top prospects. Which players are the nation's top programs targeting?