The first two months of the third Advanced LIGO and Virgo observing run (2019 April-May) showed that distant gravitational-wave (GW) events can now be readily detected. Three candidate mergers containing neutron stars (NS) were reported in a span of 15 days, all likely located more than 100 Mpc away. However, distant events such as the three new NS mergers are likely to be coarsely localized, which highlights the importance of facilities and scheduling systems that enable deep observations over hundreds to thousands of square degrees to detect the electromagnetic counterparts. On 2019 May 10 02:59:39.292 UT the GW candidate S190510g was discovered and initially classified as a binary neutron star (BNS) merger with 98% probability. The GW event was localized within an area of 3462 deg$^2$, later refined to 1166 deg$^2$ (90%) at a distance of 227 ± 92 Mpc. We triggered Target-of-Opportunity observations with the Dark Energy Camera (DECam), a wide-field optical imager mounted at the prime focus of the 4 m Blanco Telescope at Cerro Tololo Inter-American Observatory in Chile. This Letter describes our DECam observations and our real-time analysis results, focusing in particular on the design and implementation of the observing strategy. Within 24 hr of the merger time, we observed 65% of the total enclosed probability of the final skymap with an observing efficiency of 94%. We identified and publicly announced 13 candidate counterparts. S190510g was reclassified 1.7 days after the merger, after our observations were completed, with a “BNS merger” probability reduced from 98% to 42% in favor of a “terrestrial classification.