Paleorecords show that the Paleocene-Eocene Thermal Maximum (PETM, ∼56 Ma) was associated with a large carbon cycle anomaly and global warming >5 K, which persisted for at least 50 kyr. Conventional carbon cycle/climate models that include a single initial carbon input pulse over ∼10 kyr fail to reproduce the long duration of the PETM without invoking additional, slow carbon release over more than 50 kyr (hereafter referred to as bleeding). However, a potential carbon source for the bleeding, as well as its release mechanism, has hitherto remained elusive. Here I present first-principle calculations of heat transfer in marine sediments which demonstrate that a bottom water temperature anomaly as generated during the PETM takes tens of thousands of years to penetrate the top few hundred meters of deep-sea sediments. While the initial temperature rise has been suggested to cause dissociation of the majority of oceanic methane hydrate within ∼10 kyr, my calculations reveal a long tail of hydrate dissociation, causing smaller but continued carbon release substantially beyond 10 kyr. In addition, I suggest that temperature-enhanced metabolic processes in marine sediments and the absence of methane hydrate deposition during the PETM contributed to prolonged carbon input during the event. Enhanced fluxes of methane over this time scale would have sustained the carbon isotope excursion and amplified long-term greenhouse warming by elevating atmospheric concentrations of steady state CH4, or in oxidized form, CO2.