The Shockers were 44 of 62 from the stripe (71 percent), including John Robert Simon's record-setting make with 44 seconds left.
Wichita State's previous record was 40 - set in 1967 and 1986.
Fred VanVleet and Zach Brown added 13 points apiece for Wichita State (6-5). The duo each made double-digit free throws - combining for 21-of-24 shooting. Baker made all seven of his free throw attempts and Shaquille Morris added 12 points.
Wichita State led 44-29 at halftime and extended to 59-35 after Baker scored eight points during a 13-2 run.