The study of capillary imbibition has ramifications in many fields, such as energy, biology, process industry, and subsurface flows. Although the capillary rise of Newtonian liquids has been the subject of several studies since the seminal works of Lucas (1918) and Washburn (1921), its generalization to the case of non-Newtonian fluids is still an open question. To fill this gap, starting from first principles, we derive a transient one-dimensional model describing the rising dynamics of shear-thinning fluid, whose viscosity is described by the Ellis viscosity model. Our model identifies the scaling for the different imbibition regimes accounting for the interplay of inertial, gravity, and viscous non-Newtonian effects (i.e., the zero-shear-rate and the shear-thinning behavior). Specifically, the rising dynamics is described by the interplay of three dimensionless parameters: the Richardson number (i.e., the ratio between potential and kinetic energies), the Ellis number (i.e., the ratio between the characteristic shear-stress of the fluid and gravity), and the shear-thinning index which quantify the degree of shear-thinning of the fluid. At early times the system follows a universal inertial regime, followed by two possible limiting regimes,i.e., the classical Lucas-Washburn and the oscillatory regimes. The competition between the governing dimensionless numbers dictates the transition between the two. We show that when the viscous effect dominates over inertia, the identification of a (time-dependent) scaling law for the effective viscosity leads to a generalization of the Lucas-Washburn theory and the rescaled trajectories toward equilibrium collapse over the classical 1/2 scaling law. On the contrary, when inertia dominates the later stage of the imbibition, the filling length oscillates around the equilibrium. By means of linear control theory, we discuss the physical mechanisms that lead to such oscillating behavior and map the different regimes in terms of the governing dimensionless parameters.