It was the first 54-hole lead for Suber, who carded a 4-under 66 on Saturday (Sunday AEST) and whose best PGA Tour finish came three weeks ago at The CJ Cup Byron Nelson when he tied for fourth.
Cauley hit his tee shot to inches on the rowdy par-3 14th hole, the start of three straight birdies that carried him to a 66, leaving him in position for his first PGA Tour title.
Missing from the mix was 36-hole leader Ben James, who plunged out of contention with a 78, and Brooks Koepka, who had to deal with a mysterious feeling in his left arm that made it difficult for him to grip the club most of the day.
"I'm struggling to grip the club with my ring finger and pinkie finger, so can't grip it," Koepka said after salvaging a 72, leaving him seven shots behind.
"My fingers would come loose, it was kind of numb. I don't know what the deal was but hopefully we'll figure it out."
He had treatment on the course and said he had never experienced anything like that.
No Australians made the cut in Canada this week. New Zealand's Ryan Fox is in a tie for seventh after three rounds.Â
James, meanwhile, watched a dream pro debut turn sour when he had five straight bogeys on the front nine, didn't make a birdie until the 15th hole and went from first to a tie for 59th.
Wyndham Clark, who won The CJ Cup Byron Nelson, shot a 63 as the leaders were just getting started. He wound up in a four-way tie for third with Tommy Fleetwood (67), Brice Garnett (67) and Jesper Svensson, who had the lead until a three-putt double bogey on the 17th dropped him to a 68.
Billy Horschel, who has been in a slump all year, shot 64 and was another shot back.
Suber has birdied the two par 5s in each of the three rounds.
"I've just hit good drives there and I feel like I've been in a good spot and iron play seems to be my strength," Suber said.
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