We prove the existence and uniqueness of variational solutions to the following non-autonomous nonclassical diffusion equation $$ u_t - \Delta u_t - \Delta u + f(u)= g(x,t) $$ in a non-cylindrical domain with the homogeneous Dirichlet boundary condition, under assumptions that the spatial domains are bounded and increase with time, the nonlinearity $f$ satisfies growth and dissipativity conditions of Sobolev type, and the external force $g$ is time-dependent. Moreover, the non-autonomous dynamical system generated by this class of solutions is shown to have a pullback attractor $\hat{\mathcal A}=\{A(t): t\in \mathbb{R}\}$.