Australian Herbert, who'd held an overnight two-shot lead after he'd equalled the lowest round ever recorded at a men's major championship on Friday with his 62, was still setting the pace halfway through his third round on Saturday (Sunday AEST).
But it all began falling apart in the run-in after Herbert had bogeyed two in a row at 15 and 16, and looked sure to drop another at the par-five 17th after he drove into thick rough, had to take a penalty drop and was left with a mercilessly tough blind shot to the green from a wispy lie on a dune from 244 yards.
He took a giant rip at the ball and, amazingly, it ended up finishing just 20ft from the pin.
"I think that's the best shot I've ever hit," Herbert confided to Sky TV course reporter, former Aussie pro Wayne Riley, who also hailed it as one of the best he'd ever witnessed.
Six-time major winner Nick Faldo enthused: "That was fun, that was entertainment - and talk about brave!"
Herbert ended up parring the hole, and though he hit another poor drive at the last and had to fire out of the rough to scrabble another battling par, the 30-year-old scraped together a one-over par 71 that leaves him joint-fourth alongside American Ryan Gerard on seven under, three behind the soaring Burns.
Burns shot a five-under 65 to lead by two shots from Korean Si Woo Kim and the man of the day, New Zealander Ryan Fox, who shot the third record-equalling 62 of the week following the landmark efforts of Herbert and Burns on Friday.
Fox had started the day in 52nd position but the son of New Zealand All Blacks' great Grant Fox ended up converting nine birdies to share the lead held overnight by Herbert, some hour-and-a-half before the Aussie even teed off.
It then proved a roller-coaster of a day for the Aussie as conditions grew a little windier and trickier as he quickly squandered the lead, dropping a shot at the third hole, before rebounding with two birdies before the turn to regain the advantage from Burns.
The whole flavour of a close-knit battle then changed in the denouement, with Burns, fresh from his battle royal with champion Wyndham Clark in the US Open, turning on the style as he birdied two more holes coming home.
Burns thrived alongside Bryson DeChambeau, who was cheered by his army of fans following his two-stroke penalty over a rules breach on Friday and showed considerable battling spirit as he crafted a 69 to reach six-under, still very much in the hunt.
It looks like a wide open Open Sunday, with the top 20 all within six shots of each other. Among them is not just Herbert but his rather more unlikely Victorian pal Cameron John, who shot a 69 to finish on four under, tied for 11th alongside a host of luminaries.
They include world No.1 Scottie Scheffler (70), who had more putting gremlins, Spanish superstar Jon Rahm (70) and former champion Xander Schauffele (66). One shot better off at five under, despite cursing a disappointing finish, is local hero Tommy Fleetwood (69).
From nowhere, though, Fox can hardly believe he's contending. He made birdies at the second, third, fifth, sixth and eighth, completing an outward nine of 29, which was just one shot more than the record-breaking 28 recorded by Herbert on Friday.
The Aucklander then made another at the 10th and, despite his only bogey of the round at the 13th, responded immediately to birdie the 14th, 16th and 17th to give himself a chance of the first ever 61 if he could birdie the last.
A poor drive into a fairway put paid to that hope, but Fox played one of the shots of the week to blast out some 180 yards, even while clipping the top edge of the sand trap, to find the 18th dance floor.
From 47ft away, he putted up short to five foot but, unlike Herbert 24 hours earlier, was able to sink his par-putt.
