Pulsar wind nebulae (PWNe) are known to accelerate electrons to very high energies (VHE), but the acceleration mechanism and site remain uncertain. The pulsar wind termination shock (TS) is a natural candidate for the acceleration site, but the toroidal geometry of the magnetic field there should render shock acceleration inoperative. We propose a solution to this apparent contradiction. We find that drift motion along the shock surface keeps either electrons or positrons in the equatorial region of the TS, where they are accelerated to VHE by the first-order Fermi mechanism. In the present work, we apply our theory to the Crab Nebula, and find that both its high-energy synchrotron emission and its inverse Compton emission can be reproduced by our model. We show that the recent observations by LHAASO of the Crab Nebula up to PeV energies allow for placing new constraints on parameters of the Crab Nebula that are still poorly known. In particular, they provide new insights into the conditions of the striped wind and the immediate downstream of the TS.