Baker's layup off an assist from Fred VanVleet with 1 minute remaining broke a 64-64 tie.
After Mike McCall and Jake Barnett each missed a 3-point try for St. Louis, Early sealed the result with a layup off another VanVleet assist with 13 seconds left for a 68-65 lead.
Baker finished with a game-high 22 points, six rebounds and four steals for the unbeaten Shockers (8-0).
Early (12 points), Chadrack Lufile (12, eight rebounds), VanVleet (10, six assists) and Tekele Cotton (10, 10 rebounds) also reached double figures in points.
Dwayne Evans provided 18 points, five rebounds and four assists for the Billikens (6-2).