Yes, the chopping blade and slicing/shredding disc are top-shelf dishwasher-safe. The work bowl, cover, and pusher are also dishwasher-safe.
Place the work bowl upside down in the dishwasher. Note where you put the sharp parts and unload carefully.
If washing by hand, hold the blade by its plastic center and move it up and down in soapy water in the bowl. Never leave blades hidden in soapy water.