Solar desalination that exploits interfacial evaporation represents a promising solution to global water scarcity. Real-world feedstocks (e.g., natural seawater and contaminated water) include oil contamination issues, raising a compelling need for desalination systems that offer anti-oil-fouling capability; however, it is still challenging to prepare oil-repellent and meanwhile water-attracting surfaces. This work demonstrates a concept of molecularly dispersing functional F and Na sites on plasma-made vertically oriented graphene nanosheets to achieve an in-air and in-water oleophobic, hydrophilic surface. The graphene architecture presents high in-air (138°) and in-water (145°) oil contact angles, with simultaneously high water affinity (0°). Such surface wettability is enabled by oleophobic, hydrophobic -CFx, and hydrophilic -COONa groups of the molecules that disperse on graphene surfaces; low-dispersion (0.439 mJ m-2) and high-polarity (95.199 mJ m-2) components of the solid surface tension; and increased surface roughness produced by graphene edges. The graphene nanostructures pump water upward by capillary action but repel oil from the surface, leading to complete in-water and in-air oil rejection and universal anti-oil-fouling capability for solar desalination. Consequently, stable solar-vapor energy efficiency of more than 85% is achieved regardless of whether the feedstock is pure or oil-contaminated water (e.g., a mixture of oil floating on water, an oil-in-water emulsion), resulting in the efficient production of clean water over several days. This outstanding performance is attributed to the universal (both in-water and in-air) oleophobic wettability, together with high light absorptance contributed by nanotraps, fast interfacial heat transfer enhanced by finlike nanostructures, and accelerated evaporation enabled by sharp graphene edges.