We present a catalog of 1.4 million photometrically selected quasar candidates in the southern hemisphere over the ~5000deg^2^ Dark Energy Survey (DES) wide survey area. We combine optical photometry from the DES second data release (DR2) with available near-infrared (NIR) and the all-sky unWISE mid-infrared photometry in the selection. We build models of quasars, galaxies, and stars with multivariate skew-t distributions in the multidimensional space of relative fluxes as functions of redshift (or color for stars) and magnitude. Our selection algorithm assigns probabilities for quasars, galaxies, and stars and simultaneously calculates photometric redshifts (photo-z) for quasar and galaxy candidates. Benchmarking on spectroscopically confirmed objects, we successfully classify (with photometry) 94.7% of quasars, 99.3% of galaxies, and 96.3% of stars when all IR bands (NIR YJHK and WISE W1W2) are available. The classification and photo-z regression success rates decrease when fewer bands are available. Our quasar (galaxy) photo-z quality, defined as the fraction of objects with the difference between the photo-z zp and the spectroscopic redshift zs, |{Delta}z|=|zs-zp|/(1+zs)<=0.1, is 92.2% (98.1%) when all IR bands are available, decreasing to 72.2% (90.0%) using optical DES data only. Our photometric quasar catalog achieves an estimated completeness of 89% and purity of 79% at r<21.5 (0.68 million quasar candidates), with reduced completeness and purity at 21.5<r<~24. Among the 1.4 million quasar candidates, 87857 have existing spectra, and 84978 (96.7%) of them are spectroscopically confirmed quasars.