The aim of this study was to evaluate the content of selected minerals and total phenolic and flavonoid content of wheat bread and bread enriched with varying amounts of carrot powder (CP) and pumpkin powder (PP). In addition, the coverage of daily requirements of selected minerals was evaluated after the consumption of 100 g of each type of bread. The research included seven types of bread: wheat bread (WB) and bread enriched with 10%, 20%, and 30% CP and PP. The vegetable powders were obtained by freeze-drying. The concentrations of minerals—sodium (Na), potassium (K), calcium (Ca), magnesium (Mg), iron (Fe), zinc (Zn), copper (Cu), and manganese (Mn)—were determined by atomic absorption spectroscopy (AAS) method. Coverage of daily requirements after consumption of a serving of bread was determined based on Recommended Daily Allowance (RDA) and Adequate Intake (AI) ratios for minerals, according to the nutritional standards for the Polish population. The addition of 10% PP led to a greater increase in Na, K, Mg, and Cu in the bread compared to CP, while for Ca, Fe, Zn, and Mn a greater increase was observed with the addition of 10% CP than PP. Among macronutrients, the greatest changes in content were recorded for Ca—the addition of 10 and 20% CP and PP resulted in increases of 66, 113, 51 and 59%. Among micronutrients, the addition of CP and PP to wheat bread caused the largest changes in Cu (46–150% increase) and Mn (25–99% increase) content. Additionally, there was a tenfold increase in total phenolic content (TPC) when 30% CP was added to the bread. Consuming 100 g of bread with CP and PP provided the greatest coverage of K (41–60%), Cu (8–17.5%), and Mn (6–17%). These findings suggest that fortification of bread with vegetable powders has the potential to be a useful method of enriching the mineral composition and increasing the proportion of selected minerals in the human diet.