A sumptuous first half saw them lead 25-16 before copping 17 straight points to start the second half and fall behind 33-25 with just 20 minutes to play on Saturday night.
But a rolling maul try for flanker Luke Reimer before a brilliant run from fullback Tom Wright put them back in front, before they survived five minutes on their line post-siren to pinch the 37-33 win.
But the Hurricanes will feel hard done by, Ardie Savea diving over the line and attempting to ground the ball with the final play of the game, only to be denied on video review after inconclusive evidence to show successful grounding.
Savea was convinced he’d got it down and one angle looked to potentially show grounding, only for the referees to stick with the on-field call.
No Australian team has progressed further in the competition than the Brumbies since 2018, and they’ll finally get the chance to exercise the demons of last season’s one-point semi-final loss to the Blues in New Zealand.
They’re off to NZ to battle the Chiefs in a semi-final in the biggest test in rugby, needing to avenge a 31-21 defeat from last month in Canberra.
The Brumbies repeatedly rejected taking straightforward penalty goals while trailing 33-30 and their gambles paid off, Wright’s pace and strength seeing him beat the line and put his side in front.
They’d looked the goods early but were behind on the hour courtesy of a stunning try from Devan Flanders, the Hurricanes flanker picking up a loose ball and running 60m to score untouched.
Territory was evenly split early but the Brumbies looked vicious anywhere near the try-line, their left-hand side slashing through for three first-half tries with centre Len Ikitau involved in two of them.
He attacked the line and put winger Ollie Sapsford through first before powering over himself, Jack Debreczeni’s effort helping them to a 25-16 halftime lead.
Kini Naholo had opened the scoring inside two minutes for the Hurricanes with a steady diet of Brett Cameron penalty goals keeping them within range at the break.
They didn’t have a ruck inside the Brumbies’ 22m zone until the 48th minute, weaponry flanker Savea quickly scoring to tighten things up at 25-23.
Five-eighth Debreczeni had his best game for the Brumbies to date and justified his selection in the starting side ahead of Wallabies prospect Noah Lolesio, repeatedly threatening the line and scoring a try to go with a strong kicking display.