Davis' countryman Luke Donald shared the lead until a bogey at his last. He carded a 5-under 66 and is tied for second place with Darren Clarke, who aced the par-3 sixth, and Brett Quigley.
Tiger Woods put himself in good position to reclaim the top spot in the world ranking. If Woods finishes better than a solo fourth place, he would overtake Vijay Singh for first in the rankings.
Woods shot a 4-under 67 and is tied for fifth place, but he could have played much better. He tallied three three-putts and missed a pair of birdie putts from inside 10 feet.
'I played like a fool today,' said Woods. 'I putted terrible. It's an absolutely horrific day on the greens. I'm pleased with the way I hit the ball, but disappointed with the way I putted today.'
This marked Woods' best start at this event, which is the only one Woods has played at least four times and never won.
The weather played havoc on Riviera and the tournament Thursday. Wind changed several times for the players in the morning tee times before a steady rain pelted the course.
Davis had the final tee time of the morning group and did not seem affected by the weather. At the par-5 first hole, Davis hit a driver, then knocked a 4-iron to 4 feet. He sank the eagle putt, then played solidly around the turn.
At the sixth, Davis converted a 5-footer for birdie, then made it two in a row with an 11-footer at No. 7. Davis parred the eighth hole, but tapped in a short birdie putt at the ninth to make the turn at 5-under 30.
'I got the ball rolling straight away,' said Davis. 'I made it close a few times. I made some good putts.'
Davis started strong on the back nine by running home a 9-footer for birdie at the 10th. Then the wind shifted and was into his face on the second nine, but Davis remained steady.
He parred every hole, but looked to be in trouble at the 15th, when Davis was looking at a bogey. The Englishman holed a clutch 16-footer for par, then parred out to get the 18-hole lead for the first time in his career, including his time on the European Tour.
'The back nine played really tough,' admitted Davis. 'I made a key par putt with four holes to go to keep my momentum going. It played tough coming in and the rain started, so I was obviously pleased to get in at 6 under par.'
Donald soared to the top of the leaderboard with a birdie at his opening hole, the 10th at Riviera, and some spectacular golf around the turn. He collected three birdies in a four-hole span from the 17th, then looked like he was going to take sole possession of the lead.
He sank two long birdie putts at four and five, then hit his tee ball inside 2 feet to set up birdie at the sixth and join Davis at minus-6. Donald's drive at nine landed in a divot and his approach found sand. He made bogey to fall into a tie for second.
'You've got to take the good with the bad,' said Donald. 'The rain wasn't coming down too hard and it didn't feel like it played that difficult out there.'
Clarke was 3 under on his round thanks to four birdies and one bogey. Then he came to the 199-yard, par-3 sixth. He hit a 7-iron that pitched over the flag and watched as it trickled back into the hole for the only ace on Thursday.
'A little bit of skill and a lot of luck,' said Clarke, referring to his hole-in-one.
Quigley birdied his first four holes then added three more birdies and two bogeys for his 66.
Two-time defending champion Mike Weir is part of the group tied with Woods in fifth place. Jose Coceres, Adam Scott, Kevin Sutherland, Omar Uresti and James Driscoll are knotted with Woods and Weir two shots out of the lead.