Following our discovery of radio pulsations from the newly recognized anomalous X-ray pulsar (AXP) 1E 1547.0 - 5408, we initiated X-ray monitoring with the Swift X-ray telescope and obtained a single target-of-opportunity observation with the Newton X-ray Multi-Mirror Mission (XMM-Newton). In comparison with its historic minimum flux of 3 x 10(-13) ergs cm(-2) s(-1), the source was found to be in a record high state, f(X)(1-8 keV) = 5 x 10(-12) ergs cm(-2) s(-1), or L-X = 1.7 x 10(35)(d/9 kpc)(2) ergs s(-1), and declining by 25% in 1 month. Extrapolating the decay, we bound the total energy in this outburst to 10(42) ergs < E < 10(43) ergs. The spectra ( fitted with a Comptonized blackbody) show that an increase in the temperature and area of a hot region, to 0.5 keV and similar to 16% of the surface area of the neutron star, respectively, are primarily responsible for its increase in luminosity. The energy, spectrum, and timescale of decay are consistent with a deep crustal heating event, similar to an interpretation of the X-ray turn-on of the transient AXP XTE J1810 - 197. Simultaneous with the 4.6 hr XMM-Newton observation, we observed at 6.4 GHz with the Parkes telescope, measuring the phase relationship of the radio and X-ray pulse. The X-ray pulsed fraction of 1E 1547.0 - 5408 is only similar to 7%, while its radio pulse is relatively broad for such a slow pulsar, which may indicate a nearly aligned rotator. As also inferred from the transient behavior of XTE J1810 - 197, the only other AXP known to emit in the radio, the magnetic field rearrangement responsible for this X-ray outburst of 1E 1547.0 - 5408 is probably the cause of its radio turn-on.