Appleby, the first-round leader, started on the back nine and moved to 11 under with a birdie on No. 6. He was one of 44 players who had to return to the course early Saturday to finish the round.
Owen teed off late in the afternoon and birdied five of his last seven holes for a 65, the lowest score of the tournament.
A month ago, a three-putt from 3 feet cost Owen a chance to win at Bay Hill. Owen talked to noted sports psychologist Bob Rotella shortly after the gaffe.
'Concentrate on everything you did right,' Owen said. 'The majority of the things were all positive.'
The 34-year-old Englishman had plenty of those on Friday, too. He made a 3-foot birdie putt on 13, holed a bunker shot on the par-3 14th and finished the round with a bending 20-footer on the difficult 18th.
'I'm just trying to have a good time and I'm playing pretty well, which helps,' Owen said. 'Just trying to see the good shots more than the bad shots.'
Appleby started his round a half-hour later. Like Owen, Appleby birdied the two par 5s on the back nine. The Australian tacked on two more birdies as the sun was setting to retake the outright lead.
'You put it in the fairway. The game becomes a lot easier,' Appleby said. 'I drove it well, putted pretty good. But nothing fancy. Just good, solid golf.'
South Africa's Trevor Immelman was 8 under after a 67.
The 26-year-old Immelman has missed the cut in his last four starts, but said his results don't reflect how well he's played.
'I would say frustration sums the whole thing up because I really felt like the whole year I've hit the ball quite well,' he said.
A victory in Texas just might top his experience at the Presidents Cup last September, what he termed the 'best golfing week of my life.'
Immelman left wanting to emulate the players' demeanors as much as their games.
'When you look at the top players, there's one common denominator and that's how relaxed they are,' he said. 'There's no drama, no real massive emotional changes. Those guys are pretty constant, they're level, they believe in themselves and they get the job done.'
Mathias Gronberg was 7 under after a 69. He shot a 68 in blustery conditions on Thursday and was disappointed he didn't go lower in the second round.
'I'm amazed no one really shot lower than 5 under in the morning,' Gronberg said. 'It was really there to shoot a low score today. We had perfect conditions.'
Gronberg was flawless with his irons, hitting all 18 greens in regulation -- a rarity, even for pros.
'I was talking with my playing partners about how rarely we do it,' Gronberg said. 'We're supposed to be professionals, but at least for me, it's like maybe a year in between or two years since I did 18 greens.'
Australians Aaron Baddeley and Stephen Leaney were both at 6 under. Leaney had three holes to play on Saturday morning.
Two-time defending champion Vijay Singh was 4 under after a 71 that included a double bogey.
Regardless of what happens Saturday morning, Andrew Magee will miss the cut at 2 over. Magee was playing for the first time since getting a 'golf ball-size' tumor and part of his kidney removed at the Cleveland Clinic in February.
'Except for making bogeys, everything feels pretty good,' Magee said.
The 43-year-old Magee will have an MRI in August to determine if he's cancer-free. So far, doctors have given him a clean bill of health and told him chemotherapy and radiation treatment weren't necessary.
'Most of the time, you hear about friends who go through a situation like that and they have a couple years left to live,' Magee said. 'This is a happy story.'
Crowd favorite John Daly shot 31 on the front nine and a 41 on the back for a 72. Daly is 3-under for the tournament. ... Kevin Na withdrew after an opening-round 77. No reason was given. Ben Curtis withdrew later in the day. ... Former PGA champion Rich Beem was in a hurry to leave the course after shooting a 69. He was rushing to pick up his new car, a silver convertible Mini Cooper with a souped-up engine. 'The thing is a rocket,' Beem said. 'It's a silver bullet.'