Crusaders first-five Mo'unga got the nod at 10, while Hurricanes first-five Barrett starts at fullback for the Rugby Championship match at Westpac Stadium in Wellington on Saturday night.
Only Barrett, Ben Smith and Brodie Retallick have been retained in the starting XV which beat Argentina in Buenos Aires last weekend.
Smith moves to right wing to accommodate Barrett's switch to 15, while Retallick starts alongside Sam Whitelock at lock.
