Cane's intercepted pass enabled Hurricanes flyhalf Beauden Barrett to sprint clear for a counter-attacking try in the 34th minute and the back-rower then conceded an unnecessary penalty that led to Victor Vito's second half touchdown.
The Hurricanes will now host next week's final against either South Africa's Lions or last year's champions the Otago Highlanders, who clash in Johannesburg later on Saturday.
The Chiefs dominated the set piece for a majority of the contest as the Hurricanes played without influential captain Dane Coles, but explosive line speed and a ferocious defence kept the home side in the match.
Barrett steers Hurricanes into quarter-finals
They also took whatever scoring opportunities that came their way with Barrett also slotting 10 points from the boot after Willis Halaholo had opened the scoring in the sixth minute by finishing off a move set up by the flyhalf's kick and chase.
Chiefs fullback Damian McKenzie slotted three penalties for the visitors, who were unable to capitalise on numerous try-scoring opportunities.
The Hurricanes, who had been heavily penalised by referee Angus Gardner, also spent 10 second half minutes reduced to 14 men after winger Cory Jane was yellow carded for a deliberate knock down.
Neither side were really able to stamp their authority on the contest in the first quarter until McKenzie put the Chiefs on the board with a penalty soon after Halaholo's try and the hosts were then placed under siege in their own 22-metre area.
A try-saving tackle by captain TJ Perenara on McKenzie, however, relieved the pressure and the home side jumped to a 15-3 lead on Barrett's first penalty and his intercept try.
A McKenzie penalty reduced the deficit to 15-6 at the break, before Cane was penalised after he ran into Hurricanes hooker Ricky Riccitelli for no reason inside the home side's territory, that set up the field position for Vito's try.
McKenzie and Barrett then traded penalties but the Chiefs were unable to capitalise on the man advantage nor break through the yellow and black wall that blocked their path to the try line.
All Blacks' Jane, Vito, Savea dropped by Hurricanes for breaking team rules
Naholo back with two tries as Highlanders beat Chiefs