Preheat oven to 350 degrees F. Coat a 9- x 13-inch baking dish with cooking spray.
In a large bowl with an electric mixer, beat cake mix, pumpkin, eggs, and water until thoroughly combined. Pour into baking dish.
Bake 25 to 30 minutes, or until toothpick inserted in center comes out clean. Let cool. Using the handle of a wooden spoon, poke 25 to 30 holes in cake.
In a large bowl, whisk pudding mix, pumpkin pie spice, and milk until slightly thickened. Pour mixture into holes and spread evenly over top of cake. Spread whipped topping on top, cover, and refrigerate at least 4 hours or until ready to serve.