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We prove the existence of pullback attractors in various spaces for the following non-autonomous quasilinear degenerate parabolic equations involving weighted p-Laplacian operators on $\mathbb {R}^{N}$ $$ \frac{\partial u}{\partial t}-\text{div}(\sigma(x)|\nabla u|^{p-2}\nabla u)+\lambda|u|^{p-2}u+f(u)=g(x,t), $$ under a new condition concerning a variable non-negative diffusivity σ(x), an arbitrary polynomial growth order of the non-linearity f, and an exponential growth of the external force. To overcome the essential difficulty arising due to the unboundedness of the domain, the results are proved by combining the tail estimates method and the asymptotic a priori estimate method.