Preheat the oven to 350°F (175°C). Grease and flour an 8-inch (20cm) round cake pan.
In a medium bowl, whisk together the all-purpose flour, baking powder, and salt.
In a large bowl, combine the unsweetened applesauce, vegetable oil, and granulated sugar. Beat in the eggs one at a time, then stir in the vanilla extract.
Add the dry ingredients to the wet ingredients alternately with the low-fat milk, beginning and ending with the dry ingredients. Mix until just combined.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake in the preheated oven for 28 to 30 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. Then turn out onto a wire rack to cool completely before serving.