Sink drainage leaking into dishwasher inlet
The alternative to an air gap shown in J. Raefield's answer is a "high loop". 

Re-read your dishwasher installation instructions.  The drain line must be routed so that at some point between the dishwasher outlet and the sink drain connection it is higher than the fill height of the sink.  

This is to prevent sink water from entering your dishwasher plumbing, where is evidently finding the fill valve open.