The translocation of large DNA molecules through narrow pores has been examined in the context of multiscale simulations that include a full coupling of fluctuating hydrodynamic interactions, boundary effects, and molecular conformation. The actual rate constants for this process are determined for the first time, and it is shown that hydrodynamic interactions can lead to translocation rates that vary by multiple orders of magnitude when molecular weights are only changed by a factor of 10, in stark contrast to predictions from widely used free draining calculations.