We present a chemical-composition analysis of 77 red-giant stars in Omega Centauri. We have measured abundances for carbon and nitrogen, and combined our results with abundances of O, Na, La, and Fe that we determined in our previous work. Our aim is to better understand the peculiar chemical-enrichment history of this cluster by studying how the total C+N+O content varies among the different metallicity stellar groups, and among stars at different places along the Na-O anticorrelation. We find that the (anti)correlations among the light elements that would be expected on theoretical grounds for matter that has been nuclearly processed via high-temperature proton captures. The overall [(C+N+O)/Fe] increases by ~0.5 dex from [Fe/H] ~-2.0 to [Fe/H] ~-0.9. Our results provide insight into the chemical-enrichment history of the cluster, and the measured CNO variations provide important corrections for estimating the relative ages of the different stellar populations. Based on data collected at the European Southern Observatory with the FLAMES/GIRAFFE spectrograph.