I’m not entirely certain about the origins of the phrase, but I know it’s been in random use for years and years. The earliest citation in the urban dictionary is from 2003:
Winner Winner Chicken Dinner is a phrase that can be shouted out at any loud moment to signify victory or triumph. It originates from when Las Vegas casino dealers would announce customers who had won on slot machines, and said “winner winner, chicken dinner”. This means that someone has just beaten all odds by winning something big in an instance of chance (most likely gambling), which was preceded by a time where they were struggling against losing for what felt like forever.