Castagne, a defensive signing from Atalanta, made an impact at the other end by heading in a cross from fellow Belgian Dennis Praet in the 56th minute.
Vardy doubled the lead from the penalty spot in the 74th after being pulled back by Kyle Bartley.
Vardy was on target 10 minutes later again from the spot after James Justin was brought down by Dara OShea.
Vardy was the leagues top scorer last season which ended in empty stadiums. Fans are also still shut out at the start of the new campaign due to coronavirus restrictions.