The earliest ancient DNA data of modern humans from Europe dates to ∼40 thousand years ago1-4, but that from the Caucasus and the Near East to only ∼14 thousand years ago5,6, from populations who lived long after the Last Glacial Maximum (LGM) ∼26.5-19 thousand years ago7. To address this imbalance and to better understand the relationship of Europeans and Near Easterners, we report genome-wide data from two ∼26 thousand year old individuals from Dzudzuana Cave in Georgia in the Caucasus from around the beginning of the LGM. Surprisingly, the Dzudzuana population was more closely related to early agriculturalists from western Anatolia ∼8 thousand years ago8 than to the hunter-gatherers of the Caucasus from the same region of western Georgia of ∼13-10 thousand years ago5. Most of the Dzudzuana population’s ancestry was deeply related to the post-glacial western European hunter-gatherers of the ‘Villabruna cluster’3, but it also had ancestry from a lineage that had separated from the great majority of non-African populations before they separated from each other, proving that such ‘Basal Eurasians’6,9 were present in West Eurasia twice as early as previously recorded5,6. We document major population turnover in the Near East after the time of Dzudzuana, showing that the highly differentiated Holocene populations of the region6 were formed by ‘Ancient North Eurasian’3,9,10 admixture into the Caucasus and Iran and North African11,12 admixture into the Natufians of the Levant. We finally show that the Dzudzuana population contributed the majority of the ancestry of post-Ice Age people in the Near East, North Africa, and even parts of Europe, thereby becoming the largest single contributor of ancestry of all present-day West Eurasians.