Moore, who has a 10,000-point lead over Fitzgibbon, will next face Frenchwoman Johanne Defay after sending New Zealander Paige Hareb packing with 13.20 to 11.00 victory.
Defay edged Costa Rican Brisa Hennessy 10.66 to 9.54.
The other title contender Peterson thrashed Silvana Lima 13.43 to 8.54 to set up a quarter-final with Australian Nikki Van Dijk, who scored 12.40 to beat American Courtney Conologue by 0.06 points.
Meanwhile, Gilmore narrowly edged compatriot Macy Callaghan 13.07 to 12.13 to book a surf-off with American world No.5 Caroline Marks, who knocked out Australian Bronte Macauley 12.67 to 9.83.
In the men’s last 16 heats, Wade Carmichael slumped 14.60 to 9.27 against Brazilian title contender Felipe Toledo.
The world No.2’s victory over the Australian pushed the title race out to a showdown at Pipe Masters, the final event of the tour.
“I’m really happy to push the title out to Hawaii, I think it’s going to make things a lot more interesting for everyone,” Toledo said.
The title race was blown even further open after reigning world champion and leader Gabriel Medina was upset by fellow Brazilian Caio Ibelli.
Medina had been leading 14.34 to 7.17, but then in a moment of confusion started paddling his opponent’s wave without priority.
He was sensationally docked all points for his lower-scoring wave to bring the scores back to 8.17 to 7.17 and Ibelli then managed to grab a 3.10 to knock a shell-shocked Medina out.
Ibelli said there was a clear confusion over the priority as Medina doesn’t make mistake like that.
“I feel like surfing wise he won the heat, but everything went my way. I have so much respect for him,” Ibelli said. “I feel so bad, I mean I did my game and I came out with the win but he surfed pretty good.
South American Jordy Smith, Brazilian Italo Ferreira and American Kolohe Andino remain in the title race in Portugal.
American 11-times world champion Kelly Slater was edged by Japan’s Kanoa Igarashi.