Brown narrowly missed a 6-foot birdie putt for a 59, but the 60 left the 18-year-old tied at the top of the leaderboard with world number one Scheffler.
Scheffler managed a bogey-free 64 while Si Woo Kim, a past champion here, shot 65 on the tougher Stadium Course at PGA WEST and was one shot behind.
Brown made six birdies and an eagle in his first seven holes, going out in 28, and added four more birdies in his first six holes on his second nine.
"I just really like this golf course," Brown said. "There's something about the Nicklaus Course that I just really like.
"Like, it's so money. It's so pure. Everything is just perfectly well kept. I love playing with no wind. So, when you put all those things together, it's fun."
The American Express is played over three different courses - La Quinta, PGA West Stadium course and PGA West Nicklaus Tournament Course.
Collin Morikawa ended a two-year wait for a PGA Tour title with a one-shot win at the Pebble Beach Pro-Am after holding off a late surge from world number one Scottie Scheffler.
Hideki Matsuyama will take a slender one-stroke lead into the final round of the Phoenix Open on Sunday.
Scottie Scheffler kicked off his year with a four-stroke victory at The American Express in California on Sunday.
Si Woo Kim has a one-stroke lead going into the final round of The American Express on Sunday, but lurking in the chasing pack is the indomitable Scottie Scheffler.
