The Hurricanes claimed a breathless five-tries-to-two victory over the Reds, who slumped to their fifth straight loss this season.
Fullback Jordie Barrett scored 14 points from a try, three conversions and a penalty goal, while All Black brother Beauden scored the clinching try in the final minute.
Wellington led 14-0 after 19 minutes but the Reds clawed back to trail 14-12 at half-time after tries from winger Chris Kuridrani and number eight Scott Higginbotham.
The Hurricanes dominated territory and possession in the second half and had three tries disallowed by the television match official before the Barrett brothers scored tries in the final stages to wrap up victory.
Wellington have won 13 of their last 15 games against the Reds, and it was their sixth win in Brisbane.
"It's always tough coming here and the Reds have pushed most teams this season. We knew we were going to have our hands full," said skipper TJ Perenara, who scored a 80-metre intercept try in the first half.
"We started well and the Reds put us under the pump for a big period and we have to be more clinical in what we're doing and execute a bit better."
The Hurricanes took an early grip with tries by Julian Savea and Perenara.
The margin looked set to be extended to 19-0 in the 24th minute, when Beauden Barrett kicked a dropped Reds ball and won a race to it.
But not only was the try disallowed on video review, referee Angus Gardner sent inside centre Ngani Laumape to the sin bin for a charge on Jake McIntyre.
With the Hurricanes down a man, Higginbotham set up Kuridrani with an offload and then backed it up by scoring five minutes before half-time.
The Hurricanes hit back through Vince Aso to give themselves more breathing space at 19-12 but struggled to put away the Reds.
The home side showed great heart in defying the Hurricanes until the late salvo from the Barrett brothers sealed the result.
A downside to the Hurricanes victory was a likely calf injury to livewire number eight Ardie Savea in the final minutes.
The Hurricanes victory continued the New Zealand dominance of Australian opposition with a 10-0 record in this season's encounters and 32-3 in their past 35 games.