We have used the Atacama Large Millimeter/submillimeter Array (ALMA) to carry out a search for CO (3-2) or (4-3) emission from the fields of 12 high-metallicity ([M/H]...-0.72 dex) damped Lya absorbers (DLAs) at z approximate to 1.7-2.6. We detected CO emission from galaxies in the fields of five DLAs (two of which have been reported earlier), obtaining high molecular gas masses, M-mol approximate to (1.3-20.7) x (alpha C-O/4.36) x 10(10) M-circle dot. The impact parameters of the CO emitters to the QSO sight line lie in the range b similar to 5.6-100 kpc, with the three new CO detections having b less than or similar to 15 kpc. The highest CO line luminosities and inferred molecular gas masses are associated with the highest-metallicity DLAs, with [M/H] greater than or similar to -0.3 dex. The high inferred molecular gas masses may be explained by a combination of a stellar mass-metallicity relation and a high molecular gas-to-star mass ratio in high-redshift galaxies; the DLA galaxies identified by our CO searches have properties consistent with those of emission-selected samples. None of the DLA galaxies detected in CO emission were identified in earlier optical or near-IR searches and vice versa; DLA galaxies earlier identified in optical/near-IR searches were not detected in CO emission. The high ALMA CO and C[II] 158 mu m detection rate in high-z, high-metallicity DLA galaxies has revolutionized the field, allowing the identification of dusty, massive galaxies associated with high-z DLAs. The H I-absorption criterion identifying DLAs selects the entire high-z galaxy population, including dusty and UVbright galaxies, in a wide range of environments.