Zn-air batteries (ZABs) as a class of promising energy storage setups are generally powered by efficient and robust catalysts at the oxygen-involving cathode. Although the existing non-noble catalysts have outperformed noble Pt benchmark in the alkaline liquid-state ZABs, to the best of our knowledge few have excelled Pt in quasi-solid-state (QSS) ZABs. Herein, we found that an integrated Mn-Co cathode derived from the bimetallic Mn/Co metal organic frameworks generates a 1.4-fold greater power density in the QSS ZABs than a Pt cathode while its power density in liquid-state ZABs is only 0.8-fold of the latter. Moreover, such Mn-Co catalyst delivers high-rate oxygen reduction reaction (ORR) capability with half-wave potential of 0.84 V. The in-depth characterizations and analyses have demonstrated that the Co and Mn species show the specific affinity towards H₂O and O₂, respectively, synergizing the ORR process in the water-deficient environment of QSS ZABs. This work has enlightened the rational design of non-noble metal catalysts to improve the power density of QSS ZABs.