Near-infrared spectra have been obtained between 2 and 2.5 μm at six positions in Herbig-Haro object 7, which show emission lines of molecular hydrogen arising from upper energy levels that range from 6000 to 25000 K. Planar J- and C-type shock models fail to predict the relatively large columns of warm gas measured in the higher vibrational levels (v>2) of H_2. We suggest that the excess emission observed in these high-excitation lines arises from H_2 fluorescence produced by Lyalpha pumping of the lower density (~10^3-10^4) pre-shocked gas, while the bulk of the H_2 emission is excited in the hotter and denser (~10^5-10^6) post-shocked layers behind a bow C-shock. By comparing the observed line ratios with the H_2 emission from a composite `Bow C-shock+Fluorescence' model, we derive that the shock-induced UV radiation field is about 10^2-10^3 times larger than the average interstellar field, and that the ortho-to-para ratio of the fluorescent H_2 population is 1.8. The best-fitting parameters for the bow yield a shock velocity V_s=140 km s^-1, with the axis of symmetry positioned at theta=40 deg to the line of sight. This configuration suggests that the bow's dissociative cap (box apex) is producing a far-UV radiation field of the order of 0.16-1.6 erg s^-1 cm^-2.