The Australian hit seven birdies, one bogeys and one double-bogey at the Lake Karrinyup Country Club to move to 17-under and ensure a place in the last 16 on the final day along with seven others players.
Former British Open champion Louis Oosthuizen, who hit a brilliant eagle at the par-four 13, was also among the top eight after shooting a 67 to join Jason Scrivener, Austin Connelly, Adam Blyth, Lucas Herbert and Steven Jeffress at 12-under after 54 holes.
Jake Higginbottom became the sixth Australian in the top eight, taking the place on count back, after firing a 66.
After the usual three rounds of stroke play, the event will end on Sunday with 24 players competing at match play in six-hole playoffs until only the champion remains.
"You are playing an opponent one-on-one, so you have to take note of their mistakes, errors and, you know, the pressures good shots will put on you," Rumford told reporters.
"It's an internal battle but also a physical one. So yeah, it's tough match play," said the golfer whose five-shot lead would now count for nothing as the tournament switches to match play.
The top eight received byes into second round of Sunday's match play shootout with players finishing ninth to 24th facing off in the first round of the A$1.75 million ($1.35 million) tournament, tri-sanctioned by the PGA Tour of Australasia, Asian Tour and European Tour.
(Reporting by Shravanth Vijayakumar in Bengaluru, editing by Amlan Chakraborty)
