Low-mass pre-main-sequence stars in the nearby Lynds 1551 star-forming cloud are studied with the ROSAT and ASCA X-ray satellites. An 8 ks ROSAT image reveals 38 sources including seven well-known T Tauri stars, two likely new weak-lined T Tauri stars, five potential new weak-lined T Tauri stars, one young B9 star, and the remaining sources are unrelated to the cloud or poorly identified. A 40 ks ASCA image of the cloud detects seven of the ROSAT sources. Spectral fitting of the brighter X-ray-emitting stars suggests the emission is produced in either a multitemperature plasma, with temperatures near 0.2 and 1 keV, or a single-temperature plasma with low metal abundances. XZ Tau, a young classical T Tauri star, is much stronger in ASCA than ROSAT observations, showing a harder (1.5-2.0 keV) component. Timing analysis reveals all but one of the T Tauri stars are variable on timescales ranging from 1 hr to 1 year. A powerful flare, emitting 3 x 10(34) ergs within a 40 minute rise and fall, was observed by ASCA on the weak-lined T Tauri star V826 Tau. The event was preceded and followed by constant quiescent X-ray emission. The extreme classical T Tauri star XZ Tau was also caught during both high and low states, varying by a factor of 15 between the ASCA and ROSAT observations. Neither of the luminous infrared embedded protostars L1551 IRS 5 or L1551NE were detected by ROSAT or ASCA.