The increasing global energy demand and the depletion of fossil fuels have stimulated intense research on fuel cells and batteries. Oxygen electrocatalysis plays essential roles as the electrocatalytic reduction and evolution of di-oxygen are always the performance-limiting factor of those devices relying on oxygen electrochemistry. A novel perovskite with the formula of La(Co0.55Mn0.45)0.99O3-δ (LCMO) is designed from molecular orbital principles. The hydrothermally synthesized LCMO nanorods have unique structural and chemical properties and own high intrinsic activities for oxygen reduction reaction (ORR) and oxygen evolution reaction (OER). The synergic covalent coupling between LCMO and NrGO enhances the bifunctional ORR and OER activities of the novel LCMO/NrGO hybrid catalyst. The ORR activity of LCMO/NrGO is comparable with the state-of-the-art Pt/C catalyst and its OER activity is competitive to the state-of-the-art Ir/C catalyst. LCMO/NrGO generally outperforms Pt/C and Ir/C with better bifunctional ORR and OER performance and operating durability. LCMO/NrGO represents a new class of low-cost, efficient and durable electrocatalyst for fuel cells, water electrolysers and batteries.