In a medium pot, heat oil over medium-high. Add onion and cook until translucent, about 5 minutes. Add zucchini and mushrooms and cook until vegetables soften slightly, about 4 minutes. Add garlic and cook until fragrant, 30 seconds. Add tomato puree, season with salt and pepper, and bring mixture to a boil. Reduce heat to a rapid simmer and cook until zucchini is crisp-tender, about 8 minutes. Stir in oregano.
Meanwhile, in a large pot of boiling salted water, cook pasta until al dente. Reserve 1 cup pasta water; drain pasta and add to sauce, tossing to combine and adding enough pasta water to create a sauce that coats pasta. Serve immediately.