For the development of a high-quality functional bread, it is of crucial importance to find a compromise between improving the functional value of the product and impairing its quality. Therefore, the primary aim of this work was to define the effects of carob pulp powder (CP; 10–20% on flour basis) on the rheological behavior of dough and quality parameters of bread. The introduction of CP caused a major increase in dough resistance to extension while significantly decreasing dough extensibility, resulting in a lower volume with a denser crumb structure of the obtained bread samples. The negative effects of the CP on the dough and bread characteristics were partially compensated by the addition of dry sourdough (DS) as an improver at the level of 5–10% (flour basis). It was determined that by combining 15% CP and 10% DS, bread of very good sensory quality can be produced. The content of total dietary fiber (6.48 g), Mn (0.76 mg) and Cu (0.21 mg) per 100 g of the mentioned bread was sufficient for the nutritional claims “high in fiber”, “high in manganese” and “source of copper”. The obtained bread had a content of total phenols 10 times higher than the control and 5.5, 12 and 26 times higher antioxidant activity determined by FRAP, DPPH and ABTS tests, respectively. Moreover, microbiological tests confirmed the durability of the bread with 15% CP and 10% DS of up to seven days. Therefore, carob pulp powder can be successfully applied in the production of bread whose characteristics are adapted to modern trends in nutrition.