Preheat the oven to 350F degrees and adjust the oven rack to the middle position. Line a 9x13-Inch baking pan with foil - long enough so it hangs over the sides of the pan. This will make getting the cake out of the pan easier. Lightly grease the foil.
In a liquid measuring cup, add the applesauce, sour cream, and vanilla. Mix until combined and smooth. Set aside.
In a small mixing bowl, whisk together the flour, baking powder, baking soda, and salt. Set aside.
In the work bowl of a stand mixer fitted with the paddle attachment, add the butter and sugar. Mix on medium high speed to cream the butter and sugar together until light and fluffy. Add the eggs, one at a time, and mix to incorporate after each addition. Scrape down the sides of the bowl as needed.
Turn the speed to low, sprinkle in half of the dry ingredients into the stand mixer. Mix on low speed until just incorporated. Then add the applesauce mixture. Mix until just incorporated. Sprinkle in the remaining dry ingredients and mix until no dry streaks remain. Do not over mix - otherwise the cake will turn out tough.
Pour half of the batter into the prepared baking dish and spread out evenly. Sprinkle about 1/3 of the streusel topping over the batter. Pour the remaining batter over the top - carefully spreading the batter out as necessary. Then finish with sprinkling the remaining streusel mix on top.
Bake the cake for 30 to 40 minutes, or until the cake feels firm and a toothpick inserted in the middle comes out mostly clean. Allow the cake to cool in the pan set on a wire cooling rack for 30 minutes. Lift the cake out of the pan using the edges of the foil, and let cool completely on the wire rack.