Preheat oven to 350 degrees F (175 degrees C). Lightly grease a 9x5-inch loaf pan.
Combine flour, baking soda, and salt in a large bowl. Beat applesauce, yogurt, and brown sugar with an electric mixer in a large bowl until light and fluffy. The mixture should be noticeably lighter in color. Add room-temperature eggs one at a time, allowing each egg to blend into butter mixture before adding the next. Stir in mashed bananas until well blended. Pour banana mixture to flour mixture; stir just to moisten. Transfer batter into prepared loaf pan.
Bake in preheated oven until a toothpick inserted into center of loaf comes out clean, 60 to 65 minutes. Let bread cool in pan for 10 minutes, then turn out onto wire rack to cool completely.