The use of natural gas (NG) as supplement of the normal diesel fuel in compression ignition (CI) environments (Natural Gas/Diesel Dual-Fuel, NG/DDF), seems to present an answer towards reducing soot or particulate matter (PM) and nitrogen oxides (NOx) emissions in existing and future diesel engine vehicles. The benefits for the environment can be even higher, as recently NG quality gas can be produced from biomass (bio-methane or bio-CNG or ‘green gas’). However, this engine type where the main fuel is the gaseous one and the diesel liquid fuel constitutes the ignition source (pilot), experiences higher specific energy consumption (SEC), carbon monoxide (CO), and unburned hydrocarbons (HC) emissions compared to the conventional (normal) diesel one, with these adverse effects becoming more apparent under partial load operation conditions. Apart from using bio-fuels as pilot fuel, it is anticipated that air oxygen enrichment—addition of oxygen in the intake air—can mitigate (at least partly) the associated negative results, by accelerating the burning rate and reducing the ignition delay. Therefore, the present work strives to investigate the effects of various degrees of oxygen enrichment on the combustion, performance, and emissions of such a NG/DDF engine, operated under various loads and pilot (diesel fuel) quantities. The study is carried out by using an in-house, comprehensive, computational model, which is a two-zone (phenomenological) one. The accuracy of the modeling results are tested by using related experimental data from the literature, acquired in an experimental investigation conducted on a naturally aspirated, light-duty, NG/DDF engine. The computational study is extended to include various pilot fuel quantities, attempting to identify the influence of the examined parameters and witness advantages and disadvantages. The study results demonstrate that the air oxygen enrichment reduces the specific energy consumption and CO emissions, by accelerating the burning rate and reducing the ignition delay (as revealed by the cylinder pressure and rate of heat release diagrams), without impairing seriously the soot and NO emissions. The conclusions of the specific investigation are much useful, particularly if wished to identify the optimum combination of the parameters under examination for improving the overall performance of existing CI engines functioning under natural gas/diesel fuel operating mode.