Browne, never a winner in his three years of competition since turning 50, set the tournament record for lowest score through 54 holes. After a record-tying 64 followed by 69, he's at 15-under 198.
The journeyman, whose longtime caddie is Augusta's Buck Moore, had all the answers for the third day in a row.
"The truth is, I haven't been thinking about what I've been doing; I'm thinking about what I'm going to do," Browne said.
O'Meara, winner of the 1998 Masters Tournament and British Open, had 66.
Playing in the same pairing with Browne, O'Meara pulled even with Browne with a birdie at the ninth hole but Browne had five birdies on the difficult back nine.
"I've always been a player that respects my fellow players," O'Meara said. "Olin's a friend. He's a very good player. He showed that out there today and I know he'll be tough tomorrow."
Heading into the final round, it looks to be a two-man race. Six shots off the pace at 9-under 204 are Jeff Sluman (65), Peter Senior (68), Joey Sindelar (69) and Mark Calcavecchia (69).
Browne is trying to join Simon Hobday, who won at Pinehurst No. 2 in 1994, as the only wire-to-wire winners in the tournament's 32 years.
Browne began the day with a one-shot edge on O'Meara. They matched each other swing for swing, putt for putt, throughout a day of scorching heat and high humidity at Inverness Club.
Both birdied the fourth hole, with O'Meara falling two back with a bogey on the next hole. Birdies on holes 8 and 9 helped O'Meara catch Browne heading to the back nine.
Browne hit a wedge to 18 inches for birdie on the 10th hole and took off. He built a three-stroke lead with two more birdies through 13.
O'Meara birdied the 17th and 18th to pull within a shot, but each time Browne dropped in a birdie putt immediately on top of O'Meara's.
"Does it feel good to answer his birdie? Yeah, of course it does," Browne said. "But I'm not looking at it that way. This isn't match play."
Inverness again remained defenseless despite sun and high temperatures. The course was hit with almost 4 inches of rain in the last week, with a huge thunderstorm on Friday morning delaying play and forcing 21 players to return to the course on Saturday morning to complete the second round.
Barely staying in range of Browne and O'Meara are Hale Irwin, who shot his age (66), along with New Mexico club pro Jeff Roth (68) and Michael Allen (71), all at 7-under 206.
Tied for 10th and nine shots back came a who's who of senior golf, including defending champion Bernhard Langer (68), Hal Sutton (66), Nick Price (68), Jay Haas (68), Steve Jones (69) and Corey Pavin (70).