Season the short ribs with salt and pepper. In a large skillet, heat the oil over medium-high heat. Working in batches, add the ribs and cook, turning often, until browned, 7 to 10 minutes. Transfer to a large slow cooker.
Pour off all but about 1 tablespoon fat from the skillet. Add the onion and garlic and cook, scraping up any browned bits, until slightly softened, about 4 minutes. Stir in the wine and tomatoes and bring to a boil. Transfer the mixture to the slow cooker and cook on low heat until tender, about 6 hours.
Transfer the ribs to a platter and cover to keep warm. Strain the cooking liquid into a large measuring cup and skim as much fat as possible from the surface. Transfer to a saucepan, bring to a boil and cook until reduced to about 1 cup. Whisk in the tomato paste and season with salt and pepper. Spoon the sauce over the ribs.