There was also an upset loss for the Queensland Reds, who went down 36-33 to bottom club Fijian Drua in Suva, but the third-placed ACT Brumbies closed on the leaders with a 40-17 win over the NSW Waratahs in Canberra.
The Chiefs still top the standings with the Canterbury Crusaders, who had a bye and will host a top-of-the-table clash between the teams next week, but the loss to the Hurricanes allowed the Brumbies to cut the gap to three points.
The Hurricanes topped the standings at the end of the last regular season but made a poor start this year and at one stage looked set to miss out on the six-team playoffs altogether.
A switch to flyhalf from the outside backs for Ruben Love helped trigger something of a revival and they went into Saturday's clash with the Chiefs at Wellington Stadium on the back of an impressive away win over the Brumbies.
The Chiefs held a 17-9 lead at halftime on the back of two tries from Daniel Rona but Love, Callum Harkin and Billy Proctor produced some backline magic in the second half and Sullivan, an early replacement for Kini Naholo, cashed in.
"We're rugby players so sometimes we're a bit slow to learn and slow to take things on," captain Du'Plessis Kirifi said, explaining the turnaround in the Hurricanes' season.
"But we're picking it up now and the boys are willing and eager to learn and extremely coachable. So great job this week and we're not done yet."
The win moved the Hurricanes up to fifth place in the standings, two points clear of Moana Pasifika in sixth.
Moana also had a replacement to thank for their first ever win over the Otago Highlanders in a Dunedin thriller in the last match of the round on Sunday.
The Highlanders looked like taking the win when Jack Taylor scored off the back of the rolling maul in the 73rd minute, but Samoa scrumhalf Melani Matavao charged down a clearance kick after the restart and touched down to give Moana a 34-29 win.
"We want to win the championship, but first we've got to make the top six, which means winning every week," said inspirational Moana skipper Ardie Savea.
"We take the W today, we go home, hug our loved ones ... reset away from footy, and then we come back next week and play the Blues at home."
The Auckland Blues continued their mini revival at Eden Park on Friday night with a 40-19 bonus point win over Western Force orchestrated by twice World Player of the Year Beauden Barrett at flyhalf.
Force fullback and former Wallaby Kurtley Beale scored an early try on his long-awaited return from injury but the Perth-based outfit were unable to contain the reigning champions in the second half.
The Force dropped to eighth behind the seventh-placed Blues on points difference but a point ahead of the Waratahs, who slumped to an eighth successive loss to the Brumbies in the Australian capital on Saturday night.
Fullback Joseph Suaalii again shone for the Waratahs but tries from prop Allan Alaalatoa, hooker Billy Pollard and flying winger Corey Toole in the third quarter saw the Brumbies safely home.