Ryan Fox rocketed into contention with the third record-equalling 62 in 24 hours on the bone-dry links before Burns emerged from an attritional afternoon to seize the lead.
US Open runner-up Burns followed his sizzling 62 on Friday with a five-under 65 to reach 10 under heading into Sunday's climax when he will try to bag his first major.
New Zealand's Fox was on eight under along with South Korean Kim Si-woo while American Ryan Gerard and Australian halfway leader Lucas Herbert were on seven under.
DeChambeau, who was punished after an inadvertent breach of R&A rules late on Friday following a second-round 66, again showed his new-found links course discipline although a bogey at the 18th left him four strokes behind playing partner Burns.
Putting gremlins
World number one Scottie Scheffler suffered a frustrating day and made only one birdie in a level-par 70 that left the defending champion six shots off the pace.
His putting gremlins continued and unless he does something incredible on Sunday the Claret Jug is likely to be lifted by a 13th successive first-time champion.
"A little bit frustrating, but if I hole some putts tomorrow, I could shoot a really low round and move my way up the leaderboard," Scheffler said.
Local favourite Tommy Fleetwood led the home challenge in front of packed galleries as the Englishman was tied for ninth on five under after a round of 69.
Prior to the 154th Open, only five rounds as low as 62 had been carded in the history of men's majors with Branden Grace's effort at the 2017 Open, also at Birkdale, and those of Rickie Fowler and Xander Schauffele at the 2023 US Open and Schauffele and Shane Lowry at the 2024 PGA Championship.
That number has now been swelled to eight with Fox taking advantage of tranquil early conditions to emulate the feats of Burns and Herbert.
Like Herbert, he had a putt for a history-making 61 on the 18th but it did not drop.
"Standing on the last tee, I'm going if I can get one on the fairway here, you can get a wedge or a 9-iron in and have a putt for 61," Fox said. "If you execute, you can score around here. If you don't, it will bite you pretty quickly."