We present results from Chandra and XMM-Newton observations of the radio pulsar B1046-58. A high-resolution spatial analysis reveals an asymmetric pulsar wind nebula (PWN), ~6'' × 11'' in size. The combined emission from the pulsar and its PWN is faint, with a best-fit power-law photon index of Γ = 1.7 and unabsorbed luminosity of ~10³² ergs s^-1 in the 0.5-10.0 keV range (assuming a distance of 2.7 kpc). A spatially resolved imaging analysis suggests the presence of softer emission from the pulsar. No pulsations are detected from PSR B1046-58; assuming a worst-case sinusoidal pulse profile, we derive a 3 σ upper limit for the pulsed fraction in the 0.5-10.0 keV range of 53%. Extended PWN emission is seen within 2'' of the pulsar; the additional structures are highly asymmetric and extend predominantly to the southeast. We discuss the emission from the PWN as resulting from material downstream of the wind termination shock, as outflow from the pulsar, or as structures confined by a high space velocity. The first two interpretations imply equipartition fields in the observed structures of 40-100 μG, while the latter case implies a velocity for the pulsar of 190 n km s^-1 (where n₀ is the ambient number density in units of cm^-3). No emission from an associated supernova remnant is detected.