Bottom currents play a major role in deep‐sea sedimentation, but their significance in the burial of organic carbon is poorly quantified at a global scale. Here we show that Holocene fluxes of organic carbon into the contourite drifts are high, with a global average of 0.09 g cm−2 Kyr−1. At individual drift sites, fluxes are commonly 1–2 orders of magnitude greater than rates in surrounding areas and in global depth‐similar zones. These high fluxes of organic carbon into the contourite drifts are due to high rates of sedimentation. Over the past 50 million years, sedimentation rates at the studied contourite drift sites have overall increased, coincident with decreasing atmospheric CO2 and a cooling global climate. Our work suggests that a ramp‐up of the bottom‐current carbon pump has accelerated removal of CO2 from the atmosphere and oceanic water, thus contributing to the overall global cooling after the Eocene Thermal Maximum. Plain Language Summary: Bottom currents play a major role in deep‐sea sedimentation, but their significance in the burial of organic carbon is poorly quantified at a global scale. Here we examine data from modern contourite drifts (large‐scale, alongslope‐trending bottom‐current deposits) across the globe and show modern fluxes of organic carbon into the drifts are high, with a global average of 0.09 g cm−2 Kyr−1. At individual drift sites, fluxes are commonly 1 to 2 orders of magnitude greater than rates in surrounding areas and in global depth‐similar zones. These high fluxes of organic carbon into the drifts are due to high rates of sedimentation in these deepwater environments, which are driven primarily by vigorous bottom currents—in other words, by a bottom‐current pump that is highly efficient at burying organic carbon. Our work suggests that a ramp‐up of the bottom‐current carbon pump, attributable to progressive intensification of global ocean circulation over the past 50 million years, has accelerated removal of CO2 from the atmosphere and oceanic water, thus contributing to the global cooling after the Eocene Thermal Maximum. Sedimentary records of past organic carbon fluxes in contourite drifts over geologic time could well prove useful in informing predictions of future climate. Key Points: Modern fluxes of organic carbon into the contourite drifts are high, with a global average of 0.09 g cm−2 Kyr−1The fluxes into the drifts are commonly 1–2 orders of magnitude greater than rates in surrounding areas and in global depth‐similar zonesOver the past 50 million years, the bottom‐current pump has accelerated removal of carbon from the oceanic water [ABSTRACT FROM AUTHOR]