With potentially delayed CO2-mitigation efforts, net-negative CO2-emissions may be required to return to acceptable limits of climate warming as defined by the Paris Agreement. The ocean is an important CO2 sink under increasing atmospheric pCO2, when physico-chemical CO2-uptake dominates. The processes that govern its role under net-negative CO2-emission regimes are, however, unclear. Here we assess changes in marine CO2-uptake and storage mechanisms under a range of idealized temperature-overshoot scenarios in an Earth System model of intermediate complexity over centennial timescales. We show that while the fate of CO2 from physical-chemical uptake is very sensitive to future atmospheric boundary conditions, storage associated with the biological carbon pump continues to increase and may even dominate marine excess CO2 storage on multi-centennial time scales. Since change in biological pump carbon is strongly linked to the oxygenation status of the ocean, improved prediction of marine deoxygenation turns out to be a key to better forecast the future of the marine carbon sink on multi-centennial time scales.