Electrically driven dielectric elastomer actuators (DEAs) are susceptible to an electromechanical instability because of a positive feedback between the applied electric field and the concomitant reduction in thickness of the membrane. This paper investigates the effect of material anisotropy on the dynamic electromechanical instability of a dielectric elastomer actuator when subjected to DC and AC voltage signals. We use a computationally efficient energy based Hamiltonian approach, which relies on the energy balance at the position of maximum overshoot in an oscillation cycle, for extracting the DC dynamic instability parameters of a biaxially prestretched anisotropic DEA. Analytical solutions are obtained for static and DC dynamic instability parameters. The AC dynamic instability fields are extracted by numerically integrating the differential equations of motion devised using the Euler-Lagrange equation. The trends of variation of the thickness stretch and electric field on the onset of static and dynamic pull-in instability with the material anisotropy parameter are presented. The results demonstrate a significant enhancement in the deformation and electric field on the onset of electromechanical instability with increase in the anisotropy parameter. The results of the present investigation can find potential application in the development and design of the dielectric elastomer actuator subjected to transient loading. © 2018 IOP Publishing Ltd.