Preheat oven to 325°.
Lightly spoon flour into dry measuring cups; level with a knife. Combine flour and next 6 ingredients. Set aside.
Beat sugar and eggs with a mixer at medium speed until well blended. Add applesauce, molasses, and butter; beat well.
Add flour mixture to applesauce mixture, beating just until dry ingredients are moist. Pour batter into an 8-inch square baking pan coated with cooking spray. Bake at 325° for 45 minutes or until a wooden pick inserted in center comes out clean. Cool in pan on a wire rack. Sprinkle with powdered sugar. Cut into 12 squares. Serve warm or at room temperature.