Craig Barlow (66), Joey Sindelar (68) and Carl Pettersson (70) are tied for fourth place at 8 under, one stroke ahead of six players knotted in seventh.
Appleby, who will play for the International team at next week's Presidents Cup, began the day tied for sixth place at 5 under, but quickly moved up the leaderboard with three birdies on the front nine.
The Australian then drained an 18-foot birdie putt at the 10th to move to 9 under and within one shot of the leaders.
Appleby strung together four straight pars after that before collecting three birdies in a row to reach the top of the leaderboard. His approach shot at the par-4 15th landed within a foot to set up an easy putt, and he two-putted for another birdie at 16 after reaching the green in two on the par-5 hole.
That gave Appleby sole possession of the lead, and he briefly reached 12 under with another birdie at the 17th. But the 34-year-old stumbled to his first bogey of the day at 18 to end the round at minus-11.
'Just good solid golf all day,' said Appleby, who collected the last of his six PGA Tour victories when he repeated at the Mercedes Championship in January. 'I drove the ball well, hit the middle of the club face a lot of times...and here we are.'
Beckman was tied for 16th at 3 under entering the round, but he got off to a hot start with birdies on his first three holes -- Nos. 10, 11 and 12. He stumbled to a bogey at the par-4 13th, but rebounded with three more birdies on the back nine to make the turn at minus-8.
On the front nine, the 35-year-old mixed birdies at Nos. 4 and 5 with seven pars to reach minus-10.
'I started off nice today...which kind of got me going,' said Beckman, who is seeking his second PGA win and first since 2001. 'I ended up shooting 5 under on the back nine, and I guess made a couple of birdies on the [front] and had kind of a pretty easy day.'
Smith also began his round on the back nine and reached 5 under with birdies at 11 and 15. His only bogey of the day came at the par-5 16th, and he went on to play the front nine at 5 under to tie Beckman for the clubhouse lead.
The 36-year-old, who won the Buick Classic in 2002 for his only PGA title, has finished in the top-5 just once this year -- at the U.S. Bank Championship in July. He is currently ranked 161st on the money list.
'It wouldn't take long to talk about my season so far,' said Smith. 'It's been a little disappointing. I really might as well say the last three years have been pretty disappointing for me.
'I felt like I've started playing a lot better the last couple of months, and it always has been kind of a thing with me that I need to make some putts, and I've started hitting the ball a little bit more consistently again the last couple of months and started putting a little bit better. Next thing you know, you're kind of enjoying playing golf again.'
Fan-favorite Jason Gore, who earned a 'battlefield promotion' from the Nationwide Tour last month, managed just an even-par 72 in his second round to fall from a tie for first place into a tie for seventh.
Gore was joined at minus-7 by Ben Crane (70), Jonathan Byrd (68), Rod Pampling (67), Justin Leonard (64) and Bob Burns (70).
Defending champion Vijay Singh shot a 4-under 68 in his second round and shares 19th place with seven other players. The cut fell at even par, with 65 players not making it to the weekend.