Directionally generating and controlling quantum states is critical in chiral quantum technologies. Here, we propose a scheme for the nonreciprocal generation of Bell states via cooperative dissipation in a qubit-magnon system, consisting of a superconducting transmon qubit and a yttrium iron garnet sphere subject to a bias magnetic field. The cooperative dissipation between them is induced by coupling both to a common transmission-line waveguide. Due to this cooperative dissipation, the system possesses a dark eigenstate within the single-excitation subspace. Under appropriate parameter conditions, the dark eigenstate reduces to a specific Bell state. With the assistance of a drive pulse, the system can be prepared in this Bell state (i.e., the dark eigenstate). In particular, we find that the dark eigenstate corresponds to different Bell states under positive (+B₀) and negative (-B₀) bias fields. Consequently, nonreciprocal Bell-state generation is achieved: +B₀ yields one Bell state, while -B₀ yields the other. Moreover, we define a fidelity-based nonreciprocal rate to quantify this nonreciprocity. Our work establishes cooperative dissipation as a versatile resource for nonreciprocal entangled-state generation, opening a promising route toward chiral quantum technologies.