Preheat the oven to 350°F (175°C). Grease a 9×5-inch loaf pan and set aside.
In a medium bowl, sift together the all-purpose flour, unsweetened cocoa powder, baking soda, and salt.
In a large bowl, whisk together the granulated sugar, brown sugar, mashed bananas, plain Greek yogurt, vegetable oil, eggs, and vanilla extract until well combined.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Pour the batter into the prepared loaf pan and smooth the top with a spatula.
Bake in the preheated oven for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and let it cool in the pan for 10 minutes.
Transfer the cake to a wire rack to cool completely before slicing and serving.