In mid-January 2008, EX Lup, the prototype of the small class of eruptive variables called EXors, began an extreme outburst that lasted seven months. We observed EX Lup during about 21 h with XMM-Newton, simultaneously in X-rays and UV, on August 10-11, 2008 -- a few days before the end of its 2008 outburst -- when the optical flux of EX Lup remained about 4 times above its pre-outburst level. The observed spectrum of the low-level period is dominated below ~1.5 keV by emission from a relatively cool plasma (~4.7 MK) that is lightly absorbed (NH~3.6E20 cm^-2) and above ~1.5 keV by emission from a plasma that is ~ten times hotter and affected by a photoelectric absorption that is 75 times larger. During the X-ray flare, the emission measure and the intrinsic X-ray luminosity of this absorbed plasma component is five times higher than during the low-level period. The soft X-ray spectral component is most likely associated with accretion shocks, as opposed to jet activity, given the absence of forbidden emission lines of low-excitation species (e.g., [O I]) in optical spectra of EX Lup obtained during outburst. The hard X-ray spectral component, meanwhile, is most likely associated with a smothered stellar corona. The UV emission is reminiscent of accretion events, such as those already observed with the Optical/UV Monitor from other accreting pre-main sequence stars, and is evidently dominated by emission from accretion hot spots. The large photoelectric absorption of the active stellar corona is most likely due to high-density gas above the corona in accretion funnel flows (abridged).