A bilayer nonwoven material for tissue regeneration was prepared from chitosan (CS) and hyaluronic acid (HA) by needleless electrospinning wherein 10–15 wt% (with respect to polysaccharide) polyethylene oxide was added as spinning starter. A fiber morphology study confirmed the material’s uniform defect-free structure. The roughness of the bilayer material was in the range of 1.5–3 μm, which is favorable for cell growth. Electrospinning resulted in the higher orientation of the polymer structure compared with that of corresponding films, and this finding may be related to the orientation of the polymer chains during the spinning process. These structural changes increased the intermolecular interactions. Thus, despite a high swelling degree of 1.4–2.8 g/g, the bilayer matrix maintained its shape due to the large quantity of polyelectrolyte contacts between the chains of oppositely charged polymers. The porosity of the bilayer CS–HA nonwoven material was twice lower, while the Young’s modulus and break stress were twice higher than that of a CS monolayer scaffold. Therefore, during the electrospinning of the second layer, HA may have penetrated into the pores of the CS layer, thereby increasing the polyelectrolyte contacts between the two polymers. The bilayer CS–HA scaffold exhibited good compatibility with mesenchymal stem cells. This characteristic makes the developed material promising for tissue engineering applications.