Mathias Gronberg fired a 5-under 67 to climb into second place at 12-under-par 204. Bob Estes (66) and Jerry Smith (69) share third at minus-10. K.J. Choi (67) stands alongside Trevor Immelman (71) in fifth place at 9-under-par 207.
The second and third rounds were both completed Saturday after thunderstorms on Friday delayed action for over two hours (play was eventually suspended due to darkness).
Appleby closed his second round with two pars to lead by one over Greg Owen. The Australian was joined in the lead at minus-11 when Owen birdied the first hole of the third round.
Unfortunately for the Englishman, a bogey at the third was a sign of things to come. He struggled to a 3-over 75 to tumble back to 7-under-par 209 and a tie for 13th.
Appleby got up and down from a greenside bunker for birdie on the par-5 fourth to get to 12 under, two clear of Owen. He picked up his next birdie on the sixth hole of the Tournament Course at Redstone Golf Club, where he chipped in from over the green.
'The par-5 I made a nice little 5-foot birdie putt,' Appleby said. 'That was a sneaky one down the hill. Then I chipped in from over the back of the green on six. It was a very difficult chip and it went in. That was a very fortunate shot there because I really was thinking four would be a good score to walk away with, so three was a bonus.'
The 1999 Shell Houston Open winner drained a birdie putt from over 30 feet at the seventh to grab a four-shot lead on Brett Wetterich.
Appleby three-putted for bogey on eight and could only par his next two holes. He stuck his approach within 3 feet to set up birdie on 11, and was even closer on the 12th. He dropped his second shot inside two feet for a kick-in birdie that got him to 15 under par.
The seven-time winner on the PGA TOUR parred his next five holes. Appleby missed the putting surface at the last and was unable to save par from a greenside bunker.
'Anyone in here wants the story to be about them at the end of the day,' said Appleby, who has won three of the previous four times he has led after three rounds. 'We have not really had any typical Texas winds that would keep us away from flags, so you can still be reasonably aggressive.'
Gronberg opened with seven consecutive pars. He then birdied the next three holes to move to 10 under and a tie for second, three behind Appleby.
The Swede birdied the 12th, then made an 8-footer for birdie on 13 to remain within three. Gronberg parred three straight before missing the green short and left on No. 17.
He could not save par there and slipped to 11 under. No matter, as Gronberg drained a 28-foot birdie putt at the last to get within two of the leader.
'I'm obviously very pleased with the way I played today. I had a very slow round going,' Gronberg said. 'I parred the first seven holes and didn't make a few opportunities where I could have birdied. Then I hit a great bunker shot on eight and that got my round going.'
Wetterich double-bogeyed the 18th hole to cap a round of 3-under 69. That dropped him to 8-under-par 208 and into a tie for seventh. He stands alongside Jeff Gove, Paul Goydos, Richard S. Johnson, J.L. Lewis and Wes Short.
Owen is one stroke further back and shares 13th place with Nick Watney, Jeffrey Overton and 2002 PGA champion Rich Beem.