Both are looking for their first win on the PGA Tour.
Davis made an early move, shooting a 29 on the front nine with six birdies. Thatcher overtook him with three straight birdies on the back nine and had a one-stroke lead before missing a 7-foot par putt on the 17th hole.
"I executed just really well in that stretch," Thatcher said. "It's easy to say they're easy holes, and that's what you should do, but you still have to execute."
Seventeen other players were within five strokes of the lead entering the final round.
John Rollins, James Driscoll and Stuart Appleby were two strokes back. Defending champion Fredrik Jacobson, who led after two rounds, shot an even-par 70 to remain at 9 under.
Canadian golfer Graham DeLaet shot a thrid round 70 for a share of 39th place.
Davis said his fast start was helped by the thunderstorms that pushed the finish of the second round to Saturday morning and softened up the greens.
"I didn't even realize I had shot 29," said Davis, who has been a runner-up on the PGA Tour five times. "I was just playing and everything was going right. Then obviously we hit the turn and we had the wind pick up for about three holes and it was playing tough all of the sudden."
Four of the last six winners at River Highlands have been first-time winners on the tour — Jacobson last year, Bubba Watson in 2010, Hunter Mahan in 2007 and J.J. Henry in 2006.
Davis, Thatcher and Driscoll would like to join that club.
"But it's not something that is going to help me win," said Driscoll, who followed an opening round 68 with consecutive 66s. "The only thing that is going to help me win is if I keep doing what I'm doing. But, it is hopefully a good omen."
Appleby, by contrast, is looking for his 10th tour win after making just one bogey in his round. The 41-year-old Australian said he's happy to be considered the old veteran among the young guns vying for the championship.
"It would be good to get back in the winner's circle, he said. "The only cut I've made has been shaving for a long time. I played better golf when I was a rookie out here, and that's just not right."
Jacobson recovered from a bogey on 17 to birdie the final hole and stay within striking distance as he attempts to join Phil Mickelson, the 2001 and 2002 winner, as the only players to successfully defend a title at River Highlands.
"I'm really happy being three behind going into tomorrow rather than five or six," he said. "I'll take the positives."
Watson also is in the mix, one of four golfers at 8-under par. He had a 15-foot birdie putt to go 10-under at the 17th hole, but hit the ball five feet past the hole and then missed the return shot.
"It broke about six feet, maybe a gust of wind or something, and then when I putted it back up it either bounced or broke right and it shouldn't do that, obviously, because it went one way and then the other," Watson said.
Seventy-seven players spent Saturday morning finishing their second rounds.
J.B. Holmes, who underwent brain surgery last September, finished the morning with 62, the second best round of his career. But he followed that a few hours later with a 75 and was nine strokes behind the leaders.
Camilo Villegas shot a second round 64 and followed that up Saturday afternoon with a 70. Villegas, who has three top-25 finishes this season, was four behind the leaders.
"It's nice to see my name up on the leader board," he said. "It's been a tough year and it's always good to play good."
Eighty-eight players made the cut at even par. Former UCLA star Patrick Cantlay wasn't among them. He shot a 67 on Saturday morning to finish 2 over in the first tournament of his professional career.
"I struggled on the first day and never really recovered," he said.