Fill a large pot with lightly salted water and bring to a rolling boil. Stir in spaghetti and return to a boil. Cook spaghetti uncovered, stirring occasionally, until tender yet firm to the bite, about 8 minutes. Drain well in a colander set in the sink.

Meanwhile, heat oil in a large skillet over medium heat. Stir in garlic; cook until softened, around 1 minute. Add Swiss chard; cook and stir until chard stems are tender. Use some of the pasta water to help steam the chard by covering the pan.