Three-time Western Open champion Tiger Woods posted a 67 to move into a tie for fourth place with former PGA champion Shaun Micheel at 7-under-par 206.
'It's nice to be in this position,' Furyk said. 'I wanted to go out today and play a good round.'
He collected his first birdie of the day at the par-four fourth but stumbled to a bogey at the very next hole. Furyk responded with a birdie at the par-4 seventh and played his third shot within 13 feet for a birdie at the par-5 ninth.
Furyk ran off five straight pars before he reached the green in two at the par-5 15th and drained the long eagle try to jump into the lead momentarily at 12 under. Curtis soon joined him atop the leaderboard while Furyk parred his three remaining holes.
Curtis came out of nowhere to win the British Open two years ago at Royal St. George's. He has struggled since then with only one top-10 finish, a tie for eighth at last year's Memorial.
'When you're playing bad you sometimes try to force it too much,' said Curtis, who has made just three of 15 cuts this year. 'I just went back to the basics.'
Curtis picked up a birdie at the first and added a birdie at the par-4 seventh to make the turn at 9 under. He tallied a birdie at the par-5 11th and took the lead at the par-3 14th after his tee shot stopped inside 18 feet.
He hit a bad drive at the par-5 15th but played his second back into the fairway and scrambled for a par. Curtis followed with pars at the 16th and 17th holes before his second shot to the par-4 18th stopped 6 feet from the hole.
Curtis converted the birdie try to grab a share of first heading into the final round.
'You try not to think ahead of yourself,' said Curtis. 'It should be a fun battle.'
Woods had a slow start with five straight pars but he caught fire at the par-3 sixth with the first of three consecutive birdies. The top player in the game gave a shot back at the par-3 14th but he dropped his second shot to the par-5 15th within 15 feet.
Woods rolled in the eagle try and parred the last three holes to finish within five shots of the leaders.
'I've been putting well the last two days, which I haven't done all year,' said Woods. 'This year has been kind of streaky, good and bad, nothing in between. I logged in the time and I'm starting to see some results.'
World No. 2 Vijay Singh fired a 65 and finished at 6-under-par 207 alongside 36-hole leader Chris Couch, Pat Perez and Brett Quigley. Craig Perks, Duffy Waldorf, Brian Gay and Billy Mayfair were next at 5-under-par 208.