We present X-ray observations of two young energetic radio pulsars, PSRs B1046-58 and B1610-50, and their surroundings, using archival data from the Advanced Satellite for Cosmology and Astrophysics (ASCA). The energetic pulsar PSR B1610-50 is detected in X-rays with a significance of 4.5 σ. The unabsorbed flux, estimated assuming a power-law spectrum and a neutral hydrogen column density N_H of 5 × 10²¹ cm^-2, is (2.5 ± 0.3) × 10^-13 ergs cm^-2 s^-1 in the 2-10 keV band. Pulsed emission is not detected; the pulsed fraction is less than 31% at the 90% confidence level for a 50% duty cycle. We argue that the emission is best explained as originating from a pulsar-powered synchrotron nebula. The X-ray counterpart of the pulsar is the only hard source within the 95% error region of the previously unidentified γ-ray source 3EG J1048-5840. This evidence supports the results of Kaspi et al., who suggest, in a companion paper, that PSR B1046-58 is the counterpart to 3EG J1048-5840. X-ray emission from PSR B1610-50 is not detected. Using assumptions similar to those above, the derived 3 σ upper limit for the unabsorbed 2-10 keV X-ray flux is 1.5 × 10^-13 ergs cm^-2 s^-1. We use the flux limit to estimate the pulsar's velocity to be less than ~170 km s^-1, casting doubt on a previously reported association between PSR B1610-50 and supernova remnant Kes 32. Kes 32 is detected, as is evident from the correlation between X-ray and radio emission. The ASCA images of PSR B1610-50 are dominated by mirror-scattered emission from the X-ray-bright supernova remnant RCW 103, located 33' away. We find no evidence for extended emission around either pulsar, in contrast to previous reports of large nebulae surrounding both pulsars.