Heat olive oil in a large saucepan over medium heat. Stir in onion and cook until softened, about 5 minutes. Add green beans and stir until coated in oil. Cook, stirring occasionally, for 2 to 3 minutes. Add puréed tomatoes, potatoes, salt, pepper, and sugar. Stir well, then add enough water to barely cover beans.

Cover the saucepan partially with a lid. Bring to a simmer over medium-low heat and cook gently until green beans and potatoes are soft, 25 to 30 minutes. Uncover towards the end of cooking to reduce sauce if it is too watery.