We study smoothing properties of discretizations of a linear parabolic initial boundary value problem with a possibly non-selfadjoint elliptic operator. The solution at time t > 0 of this problem, as well as its time derivatives, are in Lr for initial values in Ls even when r > s. We show that similar strong stability results hold for discrete solutions obtained by discretizing in space by linear finite elements and in time by a class of A(Ξ)-stable implicit rational multistep methods (including single step methods as a special case) with good smoothing properties, as well as for certain combinations of single step methods. Most of our results are derived from the corresponding L2-bounds, shown by semigroup techniques, together with a discrete Gagliardo-Nirenberg inequality, and generalize previously known estimates with respect to admissible problems and time discretization methods. Our techniques make it possible to obtain, e.g., supremum norm error estimates for initial data which are only required to be in L1.