In a large skillet over medium-high heat, cook beef, breaking apart pieces with a wooden spoon, until lightly browned, 5 to 6 minutes. Season with 1½ teaspoons salt and ¼ teaspoon pepper. Add onion and garlic; cook, stirring, until translucent, about 5 minutes.
Add pureed tomatoes, sugar, vinegar, tomato paste, Worcestershire, and 1½ cups water. Reduce heat to medium-low; simmer until thickened, about 20 minutes. Divide evenly among the mini buns.