Commentator Justin Marshall said Barrett was "going to be in big trouble" since the hit was late and collected Hodgman "flush on the jaw".
"There's nothing mitigating it," Marshall said. "I don't think there was any intent, but the way it transpired, it was late and he didn't do enough to avoid the collision."
The Blues scored a try through Kurt Eklund from the subsequent penalty, extending their lead to 24-10 while facing 13 players, with David Havili still in the sin bin after receiving a yellow card for a cynical foul late in the first half.
Havili soon returned, and with no specialist locking cover on the bench, Dominic Gardiner came on for Barrett 20 minutes after the red card.
Barrett will now be facing a suspension likely in line with the three-week ban handed down to teammate Shilo Klein after the hooker was sent off for a similar hit against the Highlanders a fortnight ago, one of four players who saw red during round seven of Super Rugby Pacific.