We present Hubble Space Telescope (HST) near-ultraviolet (NUV) transits of the hot Jupiter WASP-121b, acquired as part of the PanCET program. Time-series spectra during two transit events were used to measure the transmission spectra between 2280 and 3070 angstrom at a resolution of 30,000. Using HST data from 61 Space Telescope Imaging Spectrograph visits, we show that data from HST s Pointing Control System can be used to decorrelate the instrument systematic errors (jitter decorrelation), which we used to fit the WASP-121b light curves. The NUV spectra show very strong absorption features, with the NUV white light curve found to be larger than the average optical and near-infrared value at 6 sigma confidence. We identify and spectrally resolve absorption from the Mg II doublet in the planetary exosphere at a 5.9 sigma confidence level. The Mg II doublet is observed to reach altitudes of R-p1/R-star, = 0.284 +/- 0.037 for the 2796 angstrom line and 0.242 +/- 0.0431 for the 2804 A line, which exceeds the Roche lobe size as viewed in transit geometry (R-eq(RL)/R-star = 0.158). We also detect and resolve strong features of the Fe II UV1 and UV2 multiplets, and observe the lines reaching altitudes of R-p1/R-star approximate to 0.3. At these high altitudes, the atmospheric Mg II and Fe II gas is not gravitationally bound to the planet, and these ionized species may be hydrodynamically escaping or could be magnetically confined. Refractory Mg and Fe atoms at high altitudes also indicate that these species are not trapped into condensate clouds at depth, which places constraints on the deep interior temperature.