The decrease in bread consumption in the daily diet observed in recent years results from the growing awareness of consumers and the desire to promote a healthy lifestyle. The valorization of sugar by-products allows for the design of new food products intended for health-conscious consumers. The aim of this study was to assess the possibility of using sugar beet pulp (SBP), without and with molasses, in bakery recipes as part of a strategy promoting increased consumption of foods rich in bioactive compounds. The basic composition of SBP was examined, and wheat flour mixtures with their addition at 0, 1, 5, and 10% were prepared. The rheological properties of the dough with flour blend samples were determined using Mixolab®. The technological quality of the bread, its nutritional value, and its antioxidant potential were assessed. The research results indicate that fortifying bread up to 5% SBP allows for obtaining products of acceptable sensory and technological quality. The bread with 5% molasses SBP (mSBP) compared to the sample with SBP without molasses (umSBP) was characterized by greater bread volume and crumb moisture, a darker color of the crumb, a more appropriate color of the crust (golden-brown), more favorable, thin-walled pores, and a fluffier crumb. Moreover, the samples enriched with mSBP contained more total polyphenols (by approx. 40%) and showed higher antioxidant activity (by approx. 50%) than the bread with umSBP. Additionally, for bread samples with 5% or more SBP added, a nutritional claim could be made that they are a “source of fiber” (i.e., at least 3 g per 100 g of product).