Uniform, stable, one-step dyeing of textile fabrics made from a mixture of natural and synthetic fibers using traditional methods remains problematic to this day. As an alternative to traditional methods, a one-step method for dyeing mixed cotton and polyester fabrics with a new natural dye—a complex of collagen and C r 3 + and C u 2 + ions—was proposed for the first time. Dyeing of pre-prepared cotton, polyester, and cotton–polyester fabrics is carried out by dipping in an aqueous dye solution, squeezing, drying, and heat setting. The influence of the mass ratio of dye components, fixation temperature, and pH of the solution on the degree of dye fixation was determined. A degree of sorption of dyes (14–25%) from the solution onto the surface of fabrics was established, which after heat fixation decreases slightly (1–3%). During washing processes, the copper complex is almost completely washed out from polyester and blended fabrics and remains in small quantities (2.4–3.8%) on the surface of cotton fabrics. After water washing, the degree of fixation of the collagen–chrome complex in fabrics is as follows: in cotton—8–10%, in polyester—10–12%, and in cotton–polyester—8–9%. The color coordinates of cotton and cotton–polyester fabrics hardly change before and after washing, but for polyester fabric, the changes are significant. The mechanism of interaction of the complex dye with the fibers was studied using the Fourier transform infrared method, and the morphology and distribution of the dye on the surface of the samples using the SEM-EDS method. The dye binds to cellulose through ionic and coordination bonds, and to polyester through joint melting. Therefore, unlike pure polyester fabric, cotton and cotton–polyester fabrics demonstrated high color fastness to washing and light fastness. The use of production waste, the exclusion of synthetic dyes and harmful chemicals, and single-stage dyeing of the material reduce the environmental burden.