Nonequilibrium magnetization states predict non-Newtonian ferrofluid properties. It is desirable to understand the corresponding flow fields and characteristics. In this study, we derive a magnetoviscosity expression coming from the effective-field method and describing the shear-thinning non-Newtonian behavior of dilute ferrofluids with finite magnetic anisotropy. A mathematical model is developed of non-Newtonian plane flow with respect to shear and pressure driving mechanisms in the presence of an applied stationary uniform magnetic field oriented in the direction perpendicular to vorticity. The results reveal that the non-Newtonian effect tends to increase the velocity and angular velocity but to reduce the magnetization strength. Moreover, an enhanced flow rate and reduced flow drag may be obtained. The maximum non-Newtonian effect is found at a ratio of the Néel relaxation time to the Brownian relaxation time of the order of 0.1.