We present a study of discrete X-ray sources in and around the bulge-dominated, massive Sa galaxy, Sombrero (M104), based on new and archival Chandra observations with a total exposure of ~200 ks. With a detection limit of L X 1037 erg s–1 and a field of view covering a galactocentric radius of ~30 kpc (115), 383 sources are detected. Cross-correlation with Spitler et al.'s catalog of Sombrero globular clusters (GCs) identified from HST/ACS observations reveals 41 X-ray sources in GCs, presumably low-mass X-ray binaries (LMXBs). Metal-rich GCs are found to have a higher probability of hosting these LMXBs, a trend similar to that found in elliptical galaxies. On the other hand, the four most luminous GC LMXBs, with apparently super-Eddington luminosities for an accreting neutron star, are found in metal-poor GCs. We quantify the differential luminosity functions (LFs) for both the detected GC and field LMXBs, whose power-law indices (~1.1 for the GC-LF and ~1.6 for field-LF) are consistent with previous studies for elliptical galaxies. With precise sky positions of the GCs without a detected X-ray source, we further quantify, through a fluctuation analysis, the GC-LF at fainter luminosities down to 1035 erg s–1. The derived index rules out a faint-end slope flatter than 1.1 at a 2σ significance, contrary to recent findings in several elliptical galaxies and the bulge of M31. On the other hand, the 2-6 keV unresolved emission places a tight constraint on the field LF, implying a flattened index of ~1.0 below 1037 erg s–1. We also detect 101 sources in the halo of Sombrero. The presence of these sources cannot be interpreted as galactic LMXBs whose spatial distribution empirically follows the starlight. Their number is also higher than the expected number of cosmic active galactic nuclei (52 ± 11 [1σ]) whose surface density is constrained by deep X-ray surveys. We suggest that either the cosmic X-ray background is unusually high in the direction of Sombrero, or a distinct population of X-ray sources is present in the halo of Sombrero.