No. 2 seed Michigan State had a scare from Bradley but pulled away late to win 76-65 and advance to meet Minnesota in an All-Big Ten second-round game.
Big Ten Player of the Year Cassius Winston scored 26 points to lead the Spartans, who haven't made it past the first weekend of the tournament since 2015.
The Missouri Valley Conference's Braves gave Michigan State all it could handle through the first 30 minutes.
Bradley led 35-34 at the half and was still up by one point with 7 minutes to go.
Then a 9-0 spurt put the Spartans ahead 63-55 with 2½ minutes to play.
The Spartans made 25 of 26 free throws.
Elijah Childs scored 19 points to lead the Braves, who were just 3 of 12 on 3-pointers in the second half after starting 6 for 9.