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From our Secret Lives of Cepheids program, the prototype Classical Cepheid, $\delta$ Cep, is found to be an X-ray source with periodic pulsation-modulated X-ray variations. This finding complements our earlier reported phase-dependent FUV--UV emissions of the star that increase $\sim$10--20 times with highest fluxes at $\sim0.90-0.95\phi$, just prior to maximum brightness. Previously, $\delta$ Cep was found as potentially X-ray variable, using XMM-Newton observations (Engle et al. 2014). Additional phase-constrained data were secured with Chandra near X-ray emission peak, to determine if the emission and variability were pulsation-phase-specific to $\delta$ Cep and not transient or due to a possible coronally-active, cool companion. The Chandra data were combined with prior XMM-Newton observations, and very closely match the previously observed X-ray behavior. From the combined dataset, a $\sim$4$\times$ increase in X-ray flux is measured, reaching a peak $L_X$ = 1.7 $\times$ 10$^{29}$ erg s$^{-1}$ near 0.45$\phi$. The precise X-ray flux phasing with the star's pulsation indicates that the emissions arise from the Cepheid and not a companion. However, it is puzzling that maximum X-ray flux occurs $\sim$0.5$\phi$ ($\sim$3 days) later than the FUV--UV maximum. There are several other potential Cepheid X-ray detections with properties similar to $\delta$ Cep, and comparable X-ray variability is indicated for two other Cepheids: $\beta$ Dor and V473 Lyr. X-ray generating mechanisms in $\delta$ Cep and other Cepheids are discussed. If additional Cepheids are confirmed to show phased X-ray variations, then $\delta$ Cep will be the prototype of new class of pulsation-induced X-ray variables., Comment: 16 pages, 3 figures, 2 tables, accepted for publication in the Astrophysical Journal