We study numerically the relaxation of a driven elastic string in a two-dimensional pinning landscape. The relaxation of the string, initially flat, is governed by a growing length L(t) separating the short steady-state equilibrated length scales from the large length scales that keep memory of the initial condition. We find a macroscopic short time regime where relaxation is universal, both above and below the depinning threshold, different from the one expected for standard critical phenomena. Below the threshold, the zero-temperature relaxation towards the first pinned configuration provides an experimentally convenient way to access all the critical exponents of the depinning transition independently.