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Rookie winger Salesi Rayasi produced a mixed bag for the home side with two tries but also a hand in three of the tries scored by the Bulls as the Hurricanes returned to the semi-finals for a fifth successive year.
Winger Cornal Hendricks crossed twice for the three-times champion visitors, who had been forced to head back to New Zealand just a week after returning home from a four-week road trip in Australasia.
"It went down to the wire and the Bulls threw everything at us, particularly in the second half," Hurricanes captain Dane Coles said.
"We have to make sure that we nail those big moments because little mistakes can have a major impact. Just proud of the effort and we've got another week so I'm stoked."
Despite the rigours of travel, the Bulls jumped out to an 8-3 lead courtesy of fullback Warrick Gelant's 13th-minute try after Peter Umaga-Jensen rushed out of the defensive line and Jesse Kriel created an outside gap.
The Hurricanes hit back with 21 unanswered points from tries by scrumhalf TJ Perenara and wingers Ben Lam and Rayasi, who had only just come onto the field as a replacement for the injured Wes Goosen.
Rayasi, however, gifted Hendricks his first try when he failed to produce an adequate tackle on his opposite winger and the Bulls went into halftime trailing only 24-14.
The 22-year-old Rayasi, whose father Filipe played for Wellington's provincial side in the 1990s, grabbed his second try soon after the break to give the home side a 29-15 lead with about half an hour remaining.
Rayasi blotted his copybook again when he was ruled to have deliberately knocked down a Bulls pass while they were on attack. Referee Nic Berry awarded a penalty try and sent him to the sinbin.
Hendricks scored his second try while Rayasi was off the field as his captain Handre Pollard directed play towards the short-handed left wing to make the score 32-28 with 20 minutes to go.
Fullback Jordi Barrett's second penalty, however, just about gave the Hurricanes the breathing room they needed with Hendricks knocking-on in the last act of the game as the Bulls chased the converted try they needed to tie-up the scores.
"We showed some great character," Pollard said. "We had a long four-week tour, went home for a week and then came back here.
"It was always going to be tough but we fought the full 80. There are no regrets."
Argentina's Jaguares will host the other semi-final next weekend against either the Australian conference champions the ACT Brumbies or South Africa's Sharks.