The creation of innovative fibrous materials based on biodegradable semicrystalline polymers and modifying additives is an urgent scientific problem. In particular, the development of biomedical materials based on molecular complexes and biopolymers with controlled properties is of great interest. The paper suggests an approach to modifying the structure and properties of the composite materials based on poly(3-hydroxybutyrate) (PHB) obtained by the electrospinning method using molecular complexes of hemin. The introduction of 1–5 wt. % of hemin has a significant effect on the supramolecular structure, morphology and properties of PHB-based fibers. Changes in the supramolecular structure intensified with the increasing hemin concentration. On the one hand, a decrease in the fraction of the crystalline phase by 8–10% was observed. At the same time, there is a decrease in the density of the amorphous phase by 15–70%. Moreover, the addition of hemin leads to an improvement in the strength characteristics of the material: the elongation at break increased by 1.5 times, and in the tensile strength, it increased by 3 times. The antimicrobial activity of the hemin-containing composite materials against Escherichia coli and Staphylococcus aureus was confirmed. The obtained materials are proposed to be used in the creation of composite systems for regenerative medicine.