Heat oil in a deep 12-inch skillet over medium heat. Cook onion for 6 minutes, then stir in garlic for 30 seconds. Add tomatoes, water, 1/2 teaspoon salt, basil and pepper and simmer for 8 minutes.
Meanwhile, mix ricotta, squeezed spinach, Parmesan, lemon zest, half the mozzarella and remaining salt in a bowl. Divide filling evenly over the fresh pasta sheets, leaving a narrow border, and roll each sheet from a short end.
Lower the sauce to a gentle simmer. Place all 8 rolls seam-side down in one layer in the skillet, arranging them in 2 rows of 4, and spoon some sauce over their tops.
Cover tightly and simmer over low heat for 15 to 18 minutes until the fresh pasta is tender and filling reaches 165°F, spooning sauce over any exposed dry spots halfway through.
Scatter remaining mozzarella over the rolls, cover and heat for 2 minutes until melted. Remove from the heat and rest for 5 minutes, then serve 2 rolls with sauce per portion.