The 18-year-old from Tennessee just missed out on a 59 when his birdie putt at his final hole slipped by on the right side.
"I'm so stoked to have shot 60 on the PGA Tour," Brown said, acknowledging that nerves "absolutely" kicked in over the final three holes.
He had made a dazzling start on the Nicklaus Tournament course, one of three in use over the first three rounds of the event.
Brown, who made his pro debut at La Quinta last year, teed off on 10 and was eight under through his first seven holes thanks to an eagle and six birdies.
After a birdie at the second, he strung together three more at the fourth, fifth and sixth to reach 12-under for the day.
He couldn't capitalise on the par-five seventh, settling for a par after his second shot found the rough.
After a par at the eighth, he had one more chance at the 16th sub-60 round ever on the PGA Tour at the par-four ninth, where a mammoth drive and a crisp wedge left him a birdie look from inside seven feet that he couldn't convert.
"I saw my spot just a little before the hole," he said. "I got a great read off David Ford. He dripped it in there and it went right dead centre. So I was thinking, OK, maybe just a little outside right edge.
"Unfortunately it didn't go in, but I'm really happy with the outcome I had today."
It was the continuation of a whirlwind week for Brown, who finished tied for 17th in the Korn Ferry developmental tour event in the Bahamas on Wednesday, then, with a sponsor's exemption to play in La Quinta, made his way to California in time to tee it up on Thursday.
He goes into the weekend tied on 17-under 127 with Scheffler, who had eight birdies in his eight-under 64 on the Nicklaus Tournament course.
Scheffler is making his season debut after a 2025 campaign that included two major titles among his six wins.
Thrilled to have his name next to Scheffler's atop the leaderboard, Brown wasn't getting ahead of himself.
"We're halfway there. We've got a lot of golf to go," he said, adding that his next order of business would be getting some rest.
"I'm going to go take a nap," Brown said.
The leading duo were one stroke clear of South Korean Kim Si-Woo, who posted a 65 on the tougher Stadium Course for a 16-under total of 128.
It was a further stroke back to South Korea's Kim Seong-hyeon and American Matt McCarty.
American Andrew Putnam also flirted with a 59 before settling for a 12-under 60 on the La Quinta course that left him 12-under for the tournament.