Barrett was sent off in the 39th minute for the incident - Australia went on to win the 47-26.
At the SANZAAR review committee, Barrett pleaded guilty to Law 9.16: A player must not charge or knock down an opponent carrying the ball without attempting to grasp the player.
In his finding, Foul Play Review Committee Chairman Adam Casselden SC ruled the following:
"Having conducted a detailed review of all the available evidence, including all camera angles and additional evidence, including submissions from his legal representative, Stephen Cottrell, the Foul Play Review Committee upheld the Red Card under Law 9.16."
"With respect to sanction the Foul Play Review Committee deemed the act of foul play merited a mid-range entry point of 6 weeks due to World Rugby instructions that dictate any incident of foul play which results in contact with the head and/or neck must start at a mid-range level. However, taking into account mitigating factors including the Player's exemplary judicial record, his expressed remorse and the fact the Player has pleaded guilty at the first available opportunity, the Foul Play Review Committee reduced the suspension to 3 weeks."
"The Foul Play Review Committee adjourned the hearing to seek more evidence regarding the Player's proposed schedule of matches. Upon reconvening the Foul Play Review Committee was presented with extensive evidence demonstrating the Player's intended playing schedule."
"The player is therefore suspended for 3 weeks, up to and including Sunday 1 September 2019."
The ruling means Barrett will miss this weekend's Bledisloe Cup decider at Eden Park in Auckland, but he will be eligible to play the All Blacks Test against Tonga in Hamilton on 7 September.