We have detected significant X-ray emission from the direction of the young radio pulsar PSR J1105-6107 using the ASCA Observatory. The 11 sigma detection includes 460 background-subtracted source counts derived using data from all four ASCA detectors. The emission shows no evidence of pulsations; the pulsed fraction is less than 31%, at the 90% confidence level. The X-ray emission can be characterized by a power-law spectrum with photon index alpha = 1.8 +/- 0.4, for a neutral hydrogen column density Nh = 7 X 10E21 cm^-2. The unabsorbed 2 - 10 keV flux assuming the power-law model is (6.4 +/- 0.8) X 10E-13 ergs cm^-2 s^-1. The implied efficiency for conversion of spin-down luminosity to ASCA-band emission is (1.6 +/- 0.2) X 10E-3, assuming a distance of 7 kpc to the source. Within the limited statistics, the source is consistent with being unresolved. We argue that the X-rays are best explained as originating from a pulsar-powered synchrotron nebula.