Combining metal-organic frameworks with electrospinning technology, this work employs coaxial electrospun fibers decorated with ZIF-67 as precursors. The Fe3+/Co2+/Ni2+ ions in the electrospun fibers are converted into FeCoNi alloy nanoparticles which catalyze the pyrolysis of ZIF-67 on the fiber surface to generate reticulated carbon microfibers. This produces an efficacious bifunctional oxygen electrocatalyst. This new N-doped carbon hierarchical network (NCHNs) catalyst with a distinct secondary reticulated carbon structure possesses larger surface area and pore volume. Besides, the decoration of FeCoNi nanoparticles on the NCHNs provides effective active sites. With rapid mass and electron transfer between the hollow fibers, the synergistic effect between the FeCoNi nanoparticles and the layered carbon network can contribute to the high activity and kinetics of oxygen electrochemistry. At 10 mA cm−2, the FeCoNi/NCHNs exhibit an ORR half-wave potential of 0.885 V and an OER potential of 1.610 V, which exceeds that of the existing Pt/C and RuO2 catalysts. Furthermore, the FeCoNi/NCHNs-based rechargeable Zn-Air battery has an excellent power density (160.5 mV cm−2), high open-circuit voltage (1.48 V), large specific capacity (792 mAh g−1) and excellent cycling stability (>200 h). This work provides a convenient method for designing high performance and low cost catalysts.