Wash the vegetables well. Combine the minced celery, parsley, onion and garlic and sauté them until the onion is a light golden colour. In a large pot, bring the water to a boil. Add spinach, cabbage, beans, peas, tomatoes, potatoes and zucchini. Stir the onion mixture into the pot and simmer everything together for about 45 mins.
Remove the potatoes with a slotted spoon, mash them with a fork or a potato ricer and return to the minestrone, together with the pasta. Continue simmering, until the pasta is done. Remove the pot from the fire, stir in the pesto sauce and serve.