Boil water in a pot over medium high heat. Cook spinach in in water for 45 seconds until wilted. Remove spinach with a fine sieve and put in a blender. DO NOT DRAIN WATER. Return the spinach water to a boil. Add potatoes and spaghetti noodles. Cook about 6 or 7 minutes, until almost tender. Stir once or twice to keep from sticking. Add the green beans. Continue cooking until tender about 3 minutes longer.
Dip out 1 cup of the cooking liquid from the pot just before spaghetti and vegetables are done. Pour 1/ 2 cup of that liquid into the blender. Add pesto, salt and pepper. Mix until smooth. You may need stop and scrape the sides.
Drain liquid from the pot of spaghetti and vegetables, leaving ingredients in the pot. Mix in the pesto mix. Return to medium heat, cooking and stirring gently, until thickened and hot, about 1 to 2 minutes.