Rinse lentils in water and drain well; pick out and discard any foreign material. Add to 4-quart casserole or Dutch oven. Add the boiling water, salt, and garlic. Cover and place in a 350° F oven to begin baking.
Meanwhile, remove casings from sausage and cut into 1-inch chunks. Cook sausage in a wide frying pan over medium heat until well browned. Using a slotted spoon, remove sausage and stir into lentils. Discard all but 2 tablespoons of the drippings; add onion to drippings and cook, stirring, until soft. Stir onion and carrots into lentils.
Cover and bake for 45 minutes. Stir tomato sauce and vinegar into lentils and bake, uncovered, for 20 more minutes.