The Blues led 21-7 in the first half and then 33-21 with 15 minutes to play but both times copped a might Reds response in Saturday's second game at Christchurch's One NZ Stadium.
A converted Louis Werchon try on the buzzer forced extra time.
The Blues then camped in the Reds' half until captain Fraser McReight was pinged for incorrect entry to the ruck.
A frustrated McReight questioned the decision, arguing for a Blues knock-on given the ball spilled out during the play.
But it mattered little as Barrett, after a mixed night, slotted the winning points from a slight angle to move his side, at least temporarily, into second place.
"Shit happens, doesn't it," gutted Reds coach Les Kiss said, explaining the directive had been for McReight, and only McReight, to look for the pilfer.
"The truth is three or four things could have been picked out (in the decisive moment) and as a seven you have to ****ing look for the opportunities.
"The boys have an appetite and mindset to keep playing, despite the fact it got away from us. We didn't die wondering."
McReight said he would likely analyse the moment in coming days.
"As a No.7 you live for those moments ... you want to put yourself in that position," he said.
It was a third-straight win for New Zealand sides over Australian teams inside 24 hours at the venue and solidified a Super Rugby Pacific ladder now occupied from first to fourth by Kiwi sides.
The Reds (5-4) sit sixth, behind the Brumbies, despite producing some brilliance to halt the Blues' early momentum and then manufacture an extra period when the game looked gone.
Wallabies captain Harry Wilson scored a try and was heavily involved in one of his best games this season while flyhalf Harry McLaughlin-Phillips - with Test pair Carter Gordon and Tom Lynagh both sidelined - and halfback Kalani Thomas both caused problems.
Lock Lukhan Salakaia-Loto bumped off bodies all night to continue his fine season.
Blues fullback Zarn Sullivan scored a first-half double while Reds winger Tim Ryan dove to finish a spectacular try down the right edge and also produced a solid try-saving tackle on Cole Forbes.
Ryan's try made it a seven-point game in the final few minutes and they were offered one last shot when Barrett's kick for touch only made 10 metres.
The Reds mauled close to the line before Werchon burrowed over.
But the Blues, as first try scorers, were allowed to kick-off and the Reds defended stoutly for three minutes before the inevitable penalty.
"The boys said it was one of the toughest games they've played in a long time," Blues coach Vern Cotter said.
"Tight situations need clear heads and that's what we got."
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