The 18-year-old Russian winger was the 10th overall pick in the draft last weekend. He played 18 regular-season games and 25 playoff games last season for the KHL club in Chelyabinsk, his hometown.
Nichushkin had six goals and three assists in the playoffs and was the KHL's top rookie.
Terms of the deal announced Saturday weren't released.
Nichushkin scored the winning goal in the bronze medal game against Canada in the world junior championships in January. He also was on two other world junior teams with Russia, including one that took the goal medal in an under-17 tournament in 2012.