Saute garlic in hot oil in a large skillet over medium until lightly browned. Carefully stir in wine and next 5 ingredients and bring to a boil. Reduce heat to medium low and simmer for 20 minutes, stirring occasionally, until mixture thickens slightly.
Cook pasta according to package and drain. Stir together hot pasta, mozzarella cheese, and 1/3 cup Parmesan cheese in a large serving bowl, tossing to coat until cheeses start to melt. Pour tomato sauce over pasta mixture and toss to combine.
Serve with chopped fresh basil and freshly grated Parmesan if desired.