Cedric Kuakumensah scored 26 points, and Tavon Blackmon added 23 points and a career-high 10 assists for Brown (6-11, 1-3 Ivy League).
Matt Howard had 20 points to lead Penn (6-10, 0-3). Max Rothschild added 18 points and Jackson Donahue chipped in 17 for the Quakers, who shot 51 percent from the floor.
Rothschild made a jumper to give Penn its last lead, 81-80, with 3:10 remaining. Corey Daugherty answered with a layup and the Bears led 82-81. Donahue missed a go-ahead layup on the other end, and Spieth pulled in the rebound. Spieth followed with a layup and two free throws and Brown led 86-83 with 39 seconds left.
Sam Jones missed a potential game-tying 3 with 23 seconds to go.