Rumford started the third round three off the pace but was slipping down the standings at Centro Nacional de Golf after making the turn 1-under on a day when several players were going low.
But the West Australian took a more aggressive approach on the run to the clubhouse and it paid dividends, five birdies offset by two bogeys for a 68 which has him 14-under and in fifth place alone.
Dunne also posted a 68 and will begin the final round at 17-under, one ahead of Spain’s Nacho Elvira and two in front of tournament drawcard and crowd favourite Jon Rahm.
Rahm had the crowds in raptures Saturday as he charged around the course posting seven birdies and a bogey as he tries to claim his national Open for the first time.
The World umber Four has said a victory this week would mean more to him that the four titles he has already claimed in his short professional career and he relied on some brilliant golf to give himself a chance.
A stunning par at the 17th hole, where he found a brutally difficult lie over the back of the green, kept his momentum and the 24-year-old later admitted it might have been the most important hole of the day.
“The (up and down) on 17, that was huge,” he said. “That was as bad a lie as I could get. Ball below my feet, downhill lie.
“To be honest, I didn’t think I was going to be able to hit it inside 10 feet, I was just hoping to hit it somewhere where I can make it.
“To finish three or four feet under the hole, straight up the hill….I wasn’t expecting that.”
He said the par was crucial keeping him close enough to the lead to threaten tomorrow on a course where players have been going low all week.
“If I make bogey there then I don’t make that on 18 to get 15-under,” he said.
“Right now I’m two behind but three or four or five shots is too much to make up here because everyone’s making birdies so I just need to make sure I make them as well.”
Leader Dunne was at less than his best for much of the day but hung on bravely with some superb par saves.
He said after his round he liked his chances of claiming a second Tour victory tomorrow.
"It's probably the best my swing has felt, the best my body has felt like it's been moving, but I just didn't pull the shots off," he said.
"There was a lot of grinding there on the back nine but I'm happy to get in with four under.
“Probably wouldn't have taken it at the start of the day but given the shots that I hit, I'm pretty happy with it leading into tomorrow.
"Everything feels good, the game feels great, I just didn't pick shots. I hit a few shots without having a shot in mind so that's something to work on for tomorrow.
“Hopefully I'll have a good day. I'm confident about tomorrow.”
Rumford is the only Australian with a realistic chance tomorrow as both Jason noris and Marcus Fraser head to the final round 8-under and T27.
Fraser signed for a 71 Saturday while Norris had a 70, a particularly pleasing week unfolding for the popular South Australian who has struggled with his game since his breakthrough victory at last year’s Fiji International.
Jason Scrivener was the last of the Australians to make the cut and enters the final round near the tail of the field after a disappointing 75 dropped him to 2-under for the week and T70.
A SHORT GAME TIP FROM BRETT RUMFORD:
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