Bradley, inspired by a Riviera course that is one of his favorites, took only five putts over the final five holes, including a 10-footer for par on the last hole, for a 5-under 66 that assured him being in the final group.
The par was meaningful because he wanted to play Sunday with Mickelson, a mentor to him.
There's no telling what to expect in the final round. The phrase "routine par" was not part of Mickelson's vocabulary on a beautiful afternoon off Sunset Boulevard.
Mickelson played one chip well past the hole on the par-3 sixth so it would roll off the bank some 25 feet behind the cup and roll back. He made a 12-footer for his par.
The only disappointment was not taking advantage of enough birdie chances, twice missing birdie putts inside 8 feet and failing to give himself a good look on the par-5 17th when his wedge rolled off the front of the green.
Mickelson and Bradley were at 7-under 206, though this is hardly a two-man race at Riviera.
A dozen players were within four shots of the lead going into the final round, including defending champion Aaron Baddeley (66), Dustin Johnson (67) and FedEx Cup champion Bill Haas (68).
Johnson was tied for the lead until he botched the end of his round. He made bogey from the bunker on the 16th, then three-putted for bogey from just outside 3 feet on the 17th. He finished with a birdie and a small measure of redemption.
"I'm going to come out tomorrow and give it everything I've got," Johnson said.
Pat Perez still has a shot, too. He three-putted from 10 feet for bogey on the third hole, then took four putts on the next hole. His long birdie try from 60 feet on the fringe came up 5 feet short, and Perez took three more putts from there.
But he kept his patience, made a few birdies and shot 70. Perez was in the group one shot out of the lead along with Jonathan Byrd, who had a 69, and Bryce Molder, who one-putted his last eight greens for a 66.
Bradley surged into a share of the lead with an 8-foot birdie on the 16th, and a pitch to tap-in range on the next hole. From the right rough, he came up short of the 18th and chipped to 10 feet to set up the important closing par.
"That's kind of why I was excited to make that putt on the last hole. I figured that might be to get in the final group with Phil," he said.
Bradley always has looked up to Mickelson, and the four-time major champion invited him to play in one of his money games at The Players Championship. The idea is for young players to learn to play under pressure, and it paid off for Bradley when he captured the PGA Championship at the end of the year.
They have played together in a tournament only once, the opening two rounds of The Barclays.
"Nothing like this would be," Bradley said.
There was a stretch where Mickelson lived up to his "Phil the Thrill" reputation at Riviera.
He pulled his tee shot so far right on the par-3 sixth that it was headed for the ivy-covered fence until clipping a tree and dropping down. Then, instead of hitting a lofted chip that could run to the pin, he chose to chip some 25 feet past the hole, have it run up the bank and come down. It rolled 12 feet past the hole, and he made it for a par.
On the par-4 eighth, which gives players the option of two fairways, Mickelson found his own route. He blasted a tee shot so far left it went over a white fence and landed at the base of the stairs of a corporate tent. After getting a free drop, he threaded a 9-iron through five eucalyptus trees to 12 feet.
It would have been one of the more amazing birdies in his career, except he missed the putt.
He was introduced to a Bottle Brush on the 10th. That was the name of the tree between the pin and where his tee shot landed. Mickelson caught a tiny branch and didn't reach the green, but hit a skillful pitch to 3 feet for par.
And on the 15th, Mickelson blocked a tee shot into the gallery and discovered a man lying on his back, fearful of moving. The ball landed inside the hem of his shorts, and he remained still until Mickelson could remove the ball and take his free drop. He managed another par.
Haas birdied two of his last three holes and was in the group at 5-under 208, along with Johnson and Baddeley.
"I'm in a good position for tomorrow," Haas said. "I don't know what the leaders are going to do, but I'll be in striking distance."