Preheat the oven to 425 degrees. Bring a large pot of water to a boil; salt the water, add the fusilli and cook to al dente. Drain, reserving 1/2 cup of the cooking water, then return the pasta to the pot.
Meanwhile, in a medium saucepan, bring the marinara, 1 tbsp. olive oil and the vinegar to a simmer, stirring occasionally, over medium-low. Mix the sauce and reserved cooking water into the pasta. Transfer to a 1 1/2- to 2-qt. baking dish; top with the cheese. Bake until the cheese melts, about 10 minutes.
In another large pot, heat the remaining 1 tbsp. oil over medium. Add the garlic; stir until fragrant, about 1 minute. Add the spinach; toss until wilted, about 3 minutes. Serve the spinach alongside the pasta.