Its drift, there is an electric field across the depletion region, whenever an electron diffuses into the depletion region, the electric field sweeps it across the the drain, the current thus depends on the diffusion into the depeletion region rather than the electric field strength so current is relatively constant in saturation.
This is an excellent question and the last answer by anonymous is very correct. I will try to elaborate further. Now, when the channel is pinched off, this is due to an extension of the depletion region between the drain and the bulk into the channel region, thus due to the large(+ve) E field from drain to source, Electrons are swept into the n+(drain) region, thus u have drift current due to this movement! Now technically speaking, whenever you have a depletion region btw the N+ and P material, the current flow contribution is not only due to Drift, you have Diffusion contribution also but is assumed negligible compared to Drift current!