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Gather ingredients. Preheat the oven to 350 degrees F (175 degrees C). Lightly grease and flour a 9x13-inch baking pan.
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Beat butter, applesauce, and eggs together in a large bowl with an electric mixer until foamy.
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Add flour, brown sugar, cinnamon, vanilla, baking powder, baking soda, and salt; mix until well combined. Stir in apples.
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Pour batter into the prepared pan.
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Bake in the preheated oven until a toothpick inserted into the center of cake comes out clean, 25 to 30 minutes. Cool in the pan for 10 minutes.