Detailed studies of MgZnO-based metal-semiconductor-metal (MSM) solar-blind ultraviolet photodetector with different electrode structures are performed. A two-dimensional physical model is established based on the Poisson's equation and time-dependent continuity equations, which is verified by our experimental data of conventional electrode MSM detector. The steady-state characteristics and transient response of semicircular and triangular electrode MSM detectors are also investigated by this model. Compared with the conventional electrode, semicircular and triangular electrode devices exhibit a substantial improvement on the photocurrent. At a bias of 10 V, the steady-state saturated photocurrents for semicircular and triangular electrode devices are 14.69 nA and 24.37 nA respectively, corresponding to a 20.5% and 100% increase over the conventional electrode detector. Meanwhile, the transient peak photocurrents reach 31.38 nA and 52.09 nA respectively, both of which are notably larger than that of conventional device.