Only five rounds as low as 62 had ever graced a men's major but two arrived within minutes of each other on a sun-baked course as American Burns made it one of the most remarkable scoring days in Open Championship history.
Herbert, 30, leads on eight-under 132 with overnight leader Jackson Suber and fellow American Cameron Young two shots back and Burns at five under.
Northern Ireland's world number two Rory McIlroy, bidding for his second Open title, ensured he will be around for the weekend with his three-under 67 leaving him seven shots behind.
Reigning champion Scottie Scheffler was three under for the tournament midway through his second round.
Herbert, who began the day on level par before embarking on a birdie blitz, looked dejected as he shook hands with his playing partners on the 18th despite equalling the 62s by Branden Grace at the 2017 Open also at Royal Birkdale, Rickie Fowler and Xander Schauffele at the 2023 U.S. Open and Schauffele and Shane Lowry at the 2024 PGA Championship.
"It was a lot of fun, those first 12 holes I don't think I've ever played golf that well before," Herbert, who plays on the LIV Golf circuit, said.
"There was a great buzz out there and it felt like everyone wanted it to happen. I felt like I kind of let everyone down a bit missing that putt on the last."
With light winds lowering the course's guard, the early groups took full advantage.
Herbert rolled in six birdies to reach the turn in 28 shots - equalling the nine-hole Open record of Denis Durnian at Birkdale in 1983. His assault continued on the back nine with birdies at the 11th, 12th and 16th and when he set up another birdie putt on the 17th with a stunning chip even a sub-60 round looked possible.
He missed that but a par on the long par-four 18th would have secured a 61. After a wayward drive he left himself with a relatively straightforward up-and-down from in front of the green, but he faltered at the last.
Burns' 62 arrived via a very different route as he birdied the last three holes, including a chip-in birdie from a bunker on the 18th green that was met with a huge roar.
American Suber only saw a links course for the first time on Monday but led with an opening round 65. The wheels seemed to be falling off on Friday with three successive bogeys but he posted a one-under 69 to stay very much in the mix.
"I'll probably sleep pretty well tonight. I'm pretty exhausted," the 26-year-old said. "I know that in a major things are going to get challenging at some point. It's just about being patient and letting the good golf come later."
England's Matt Fitzpatrick, a pre-tournament tip, missed the cut on four over, but his younger brother Alex was leading the home charge alongside Scotland's Robert MacIntyre and England's Matt Wallace, who reached halfway on four under.
With sea breezes picking up in the afternoon, scoring toughened up for the later groups on the parched fairways.
Scheffler, bidding to become the first man since Ireland's Padraig Harrington in 2008 to retain the Claret Jug, carded a two-under 68 on Thursday but was struggling to make much headway in his second round.
The cut line is expected to be one over.