Oil-based nanofluid containing surfactant-capped magnetite nanoparticles are synthesized by a simple coprecipitation approach, and their magnetorheological properties are studied for different magnetic field strengths and volume fractions. We observe a distinct "plateau-like region" in the shear thinning viscosity curve, under an external magnetic field, possibly due to a peculiar alignment of the chains with respect to the field direction where the structure is stable against fragmentation. The observed plateau regime is reminiscent to that of kinetically arrested gel networks. Interestingly, such a plateau regime has been observed only above certain critical magnetic field when the dipolar interaction strength is much greater than the thermal energy where the aggregation becomes a nonequilibrium transport-limited process. The good collapse of specific viscosity data against Mason number for different magnetic field strengths onto a single curve suggests the dominance of hydrodynamic and magnetic forces on thermal force above a certain magnetic field strength. The observed increase in both static and dynamic yield stresses under the magnetic field confirms the formation of columnar structures that hinder the flow behavior. The hysteresis observed in the magnetic sweep experiments shows the inability of the chains to relax within the measurement time. The dynamic measurements confirm that the field-induced structures impart elastic behavior to the dispersion, which is found to increase with magnetic field and saturates at higher field strengths.