The American started the day four shots off the lead but three birdies through his opening 11 holes put him in share and when he picked up another three gains in four holes from the 13th, he led by four.
A bogey-bogey finish took some of the gloss off his round but Schauffele will head into day four at seven under as he looks for a first Rolex Series win.
Rafa Cabrera Bello is already a Rolex Series champion having won this event in 2017 and he was the closest challenger at five under, a shot clear of England's Jordan Smith and American duo Ryan Palmer and Jordan Spieth.
Schauffele arrived in Scotland as a man in form having won his last start on the PGA TOUR at the Travelers Championship after a run of four consecutive top-20 finishes.
He finished fifth in the DP World Tour Rankings in Partnership with Rolex in 2018 - the same year he won his only DP World Tour title at the WGC-HSBC Champions - and now he is looking to make history as the winner of the first event co-sanctioned by the DP World Tour and PGA TOUR.
"I don't want to think too far ahead but it would be very nice (to win)," he said. "It would be very special.
"It just proves that the game can travel and beating an incredible field, one of the best of the year. I'm just going to focus on what I can do now and not get too carried away.
"If you play really solid golf with a lead, you're going to win, that's just how it works out here.
"Going to have to go and clean a few things up on the range. I didn't really like how I finished. I just heard that's the hardest finishing hole on the PGA TOUR, so I guess that makes me feel a little bit better. But it's not good for me mentally."
Cameron Tringale started the day with a three-shot lead and while he made an eagle at the par-five third after an excellent approach to 14 feet, he could not extend his advantage as the chasing pack made their moves.
I'm just going to focus on what I can do now and not get too carried away
Bogeys on the sixth, seventh and eighth saw his lead cut to just one shot and he soon had Cabrera Bello for company.
The Spaniard had turned in 36 but an eagle at the tenth after a remarkable second to 11 feet kick-started a run of scoring.
He left himself three feet for birdie at the 11th, put his tee shot to 15 feet at the next and when he holed a 20-footer at the 13th, he was in a share of the lead.
And it would soon be a three-way tie as, after taking advantage of the par-five third, Schauffele hit a smart tee-shot into the sixth and recovered from a poor one at the 11th to get to six under.
An approach to seven feet at the 13th was followed by a 20-footer at the 15th and, with Cabrera Bello bogeying the 15th, two putts from off the green at the par-five 16th had Schauffele four ahead.
It looked like it could be turning into a procession but with the leader surrendering bogeys after missing the green at his final two holes, the door was left ajar to the field.
Spieth matched Schauffele with a 66 as he holed his second at the 15th from 108 yards for an eagle, adding to the eagle he made at the tenth after putting his second to five feet, two birdies and two bogeys.
Countryman Palmer took one shot more with four birdies and a bogey on the last, while Smith almost aced the 17th for the second day in a row, adding three more birdies and three bogeys in a 69.