This study focused on the development of a yogurt with an improved structure, texture and antioxidant activity level, by using apple pomace (AP) powder that was obtained in large quantities during the production of juices. The objective was to determine the sensory, physicochemical, textural and antioxidant characteristics of yogurt with the addition of AP powder (0.2–1.0%), during its shelf life. The physicochemical composition of AP was determined as follows: dietary fibers—62.73%, including pectin—23.12%; and the content of the antioxidant compounds in AP—total polyphenols (728.8 mg GAE/100 g DW), flavonoids (246.5 mg QE/100 g DW), tannins (63.54 mg TAE/100 g DW), carotenoids (4.93 mg/100 g DW) and the ability to inhibit the free radical (2433 µmol TE/100 g DW). AP addition reduces the yogurt fermentation time. The increase in the total dietary fiber content of up to 0.63% and in the insoluble fiber of up to 0.14% was attested in this study, as well as a significant increase in antioxidant activity, which correlated to the AP content. The addition of AP improved the textural properties of the yogurt during storage (20 days) and led to a significant reduction in syneresis. The influence of the AP content and the storage period on the textural characteristics and the overall acceptability of the yogurt samples were analyzed by the mutual information method. The AP content greatly influenced the yogurt’s quality, with the information analysis value for the overall acceptability being 0.965 bits. The analysis of the sensory and textural parameters of the yogurt during storage (1–20 days) demonstrated that samples with AP in proportions of 0.6–0.8% were evaluated with the highest score.