Heat oven to 400° F.
Heat the oil in a medium saucepan over medium-high heat. Add the garlic and cook for 1 minute. Stir in the tomatoes, jam, vinegar, 1 1/2 teaspoons of the salt, and 1/2 teaspoon of the pepper. Simmer until the sauce has thickened slightly, about 15 minutes.
Meanwhile, rinse the chicken and pat it dry with paper towels. Arrange the pieces in a 9-by-13-inch baking dish and season with the remaining salt and pepper. Pour the barbecue sauce over the chicken. Roast, spooning the sauce over the chicken once, until the chicken is cooked through, about 1 hour.
Tip: Tired of chicken? Try spooning this tangy, tomatoey barbecue sauce over pork chops, shrimp, or salmon.