Cobalt oxide (Co3O4) nanostructures decorated vertical graphene nanosheets (VGN) based hybrid electrodes are synthesized to fabricate binder-free high-performance supercapacitors. Tuning of microstructure, morphology, and stoichiometry of Co3O4 nanostructures is endeavoured by growing them in an oxygen medium using the pulsed laser deposition (PLD) technique. The variation in background oxygen pressure transformed the growth morphology of Co3O4 nanostructures from smooth film to nano-particulate caterpillar, flower, and cloud-like. Additionally, the variation in background oxygen pressure significantly influences the Co+2/Co+3 ratio. Moreover, the morphology variation led to a significant change in electrochemical active surface area (ECSA), and the electrochemical impedance characteristics varied from highly capacitive to nearly diffusive nature. These Co3O4/VGN hybrid electrodes exhibited an areal capacitance value of 32 mF/cm2 (1900 F/g) @ 20 mV/s with cyclic stability of 87 % after 3000 charge-discharge (CD) cycles in 1 M KOH electrolyte. Herein, a correlation between morphology, ECSA, and Co+2/Co+3 ratio with areal capacitance values is established. Finally, an asymmetric coin-cell device is fabricated using the Co3O4/VGN as a positive electrode and oxygenated VGN as a negative electrode, respectively. The coin-cell device exhibited a 4 mF/cm2 capacitance with energy and power densities of about 1.8 μWh/cm2 and 218.8 μW/cm2, respectively. Additionally, the coin-cell device displayed excellent cyclic stability of 92% after 5000 CD cycles. The potential utilization of this device is demonstrated by lighting an LED.