Preheat the oven to 350°F (175°C). Grease and flour an 8-inch round cake pan.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt until well combined.
In a large bowl, whisk the unsweetened applesauce, vegetable oil, and granulated sugar until smooth.
Add the eggs one at a time to the wet ingredients, beating well after each addition.
Stir in the pure vanilla extract.
Alternately add the dry ingredients and the low-fat milk to the wet mixture, beginning and ending with the dry ingredients. Mix until just combined; do not overmix.
Pour the batter evenly into the prepared cake pan and smooth the top with a spatula.
Bake in the preheated oven for 28 to 32 minutes or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes.
Turn the cake out onto a wire rack and let it cool completely before serving.