The prevailing hypothesis for lower atmospheric carbon dioxide (CO2) concentrations during glacial periods is an increased efficiency of the ocean's biological pump. However, tests of this and other hypotheses have been hampered by the difficulty to accurately quantify ocean carbon components. Here, we use an observationally constrained earth system model to precisely quantify these components and the role that different processes play in simulated glacial-interglacial CO2 variations. We find that air-sea disequilibrium greatly amplifies the effects of cooler temperatures and iron fertilization on glacial ocean carbon storage even as the efficiency of the soft-tissue biological pump decreases. These two processes, which have previously been regarded as minor, explain most of our simulated glacial CO2 drawdown, while ocean circulation and sea ice extent, hitherto considered dominant, emerge as relatively small contributors.