Barnes was in charge of the last meeting between the two sides, when Ireland won 32-22 in Wellington to claim a first series win in New Zealand last year. He was also in charge when Ireland beat the All Blacks 16-9 in Dublin five years ago.
More recently he was the referee at the MCG earlier this year when the All Blacks thrashed the Wallabies 38-7, and in Cardiff last season when Ian Foster’s side impressed with a 55-23 victory.
Kiwi Ben O’Keeffe will take charge of the France-South Africa quarter-final while Jaco Peyper (Wales v Argentina) and Mathieu Raynal (England v Fiji) will referee the other two games.
Barnes was the official for the All Blacks’ final pool play win over Uruguay, the 23rd time he has been in charge of an All Blacks test.
It will be his second time as the man in the middle for an All Blacks quarter-final, the last time being the side’s 20-18 defeat to France in Cardiff when Barnes and his assistant referees missed a forward pass in the lead-up to a French try in the second half.