The stability of acylated anthocyanins is still a new and unexplored subject of study. The changes in the contents of individual anthocyanins in colored-flesh potato tubers during processing have rarely been addressed in the literature. The aim of the present study was to determine how anthocyanin degradation and profiles are influenced in potatoes of the red-fleshed Herbie 26 variety by different methods of processing. Potato samples were divided into four categories to be analyzed, namely, raw material, potato cubes, French fries, and chips. The dried cubes, French fries, chips, semi-finished products, and finished products, obtained through laboratory processing, were examined for anthocyanin content and composition. The production process of cubes, chips, and French fries led to losses of the examined anthocyanins; however, these losses differed depending on the technological stage. The greatest losses of these compounds were determined after the final production processes, i.e., pre-frying, frying, and drying. Chip production led to the lowest losses of anthocyanins. Omitting the blanching stage in chip production allowed the retention of more anthocyanins. Pelargonidin-3-feruloylrutinoside-5-glucoside, having the highest percentage in the raw material (approximately 50%), followed by pelargonidin-3-caffeoylrutinoside-5-glucoside, proved to be the most thermally stable.