We present the discovery of molecular hydrogen (H2), including the presence of vibrationally-excited H2* in the optical spectrum of the afterglow of GRB 120815A at z = 2.36 obtained with X-shooter at the VLT. Simultaneous photometric broad-band data from GROND and X-ray observations by Swift/XRT place further constraints on the amount and nature of dust along the sightline. The galactic environment of GRB 120815A is characterized by a strong DLA with log(N(H i)/cm-2) = 21.95 ± 0.10, prominent H2 absorption in the Lyman-Werner bands (log (N(H2)/cm-2) = 20.54 ± 0.13) and thus a molecular gas fraction log f(H2) = -1.14 ± 0.15. The distance d between the absorbing neutral gas and GRB 120815A is constrained via photo-excitation modeling of fine-structure and meta-stable transitions of Fe ii and Ni ii to d = 0.5 ± 0.1 kpc. The DLA metallicity ([Zn/H] = -1.15 ± 0.12), visual extinction (AV ≲ 0.15 mag) and dust depletion ([Zn/Fe] = 1.01 ± 0.10) are intermediate between the values of well-studied, H2-deficient GRB-DLAs observed at high spectral resolution, and the approximately solar metallicity, highly-obscured and H2-rich GRB 080607 sightline. With respect to N(H i), metallicity, as well as dust-extinction and depletion, GRB 120815A is fairly representative of the average properties of GRB-DLAs. This demonstrates that molecular hydrogen is present in at least a fraction of the more typical GRB-DLAs, and H2 and H2* are probably more wide-spread among GRB-selected systems than the few examples of previous detections would suggest. Because H2* transitions are located redwards of the Lyman α absorption, H2* opens a second route for positive searches for molecular absorption also in GRB afterglows at lower redshifts and observed at lower spectral resolution. Further detections of molecular gas in GRB-DLAs would allow statistical studies, and, coupled with host follow-up and sub-mm spectroscopy, provide unprecedented insights into the process and conditions of star-formation at high redshift.