Dan Forsman also had a strong finish that helped him to a round of 9-under-par 62 at the TPC at The Canyons, a par-71 layout and one of three courses in rotation this week. Forsman was joined by Darren Clarke, Aaron Barber, Craig Barlow and John Senden in a tie for third.
Appleby, who has two runner-up finishes in each of his last two starts on the PGA Tour, started the five-day, marathon event with a pair of birdies over his first three holes at Southern Highlands Golf Club.
The Australian ran home a 30-foot putt for an eagle at the par-5 ninth to reach 4 under around the turn but it wasn't until the inward half that Appleby did his damage.
At the par-5 13th, Appleby picked up his first of six consecutive birdies to surge to the top of the leaderboard.
'I never had a finish like that,' said Appleby, a three-time winner on the PGA Tour. 'I have never finished with six birdies. I have had six birdies in a round before. I guess I ran out of holes. I wonder if I could have gone another hole.'
Flesch, who earned his first career victory on the PGA Tour at the HP Classic of New Orleans earlier this year, played the back nine first under lift, clean and place at Southern Highlands and opened with five consecutive birdies from the 10th.
He stumbled with a bogey at the par-4 15th but responded with a birdie at the par-3 17th.
Flesch continued his fine play on the front side with back-to-back birdies starting at the par-four first.
The 36-year-old drained an 18-foot putt for a birdie at the fifth and hit a sand-wedge inside eight feet for a birdie at the very next hole. Flesch then closed his round with a birdie at the par-5 ninth for his share of the lead.
'Knowing how this week is, especially without any wind you know it is going to be a shootout,' said Flesch. 'In an event like this you just have to keep the pedal to the medal.'
Barber, who could use a strong finish this week towards keeping his card for the 2003 season, birdied six of his first nine holes and added three birdies on the back nine for a 63 at Southern Highlands.
'I am really excited about the next two days to try and keep it going well,' said Barber.
Chad Campbell, who finished second at the PGA Championship, tallied six birdies and an eagle for a bogey-free round of 64 at the TPC at Summerlin.
'It was a good start. It is what I was looking for in this tournament,' said Campbell. 'You have to make a lot of birdies here if you want to have a chance to win.'
Campbell was joined by Charles Howell III, Russ Cochran, Jeff Brehaut, Bob Burns, Tim Petrovic and Rich Beem in a tie for eighth at 8 under par.
Jerry Kelly, Mark Wilson, Deane Pappas and Scott Verplank were one shot further back at minus-7.
U.S. Open champion Jim Furyk, who has had tremendous success at this event with three wins, shot a 65 at the TPC at The Canyons to finish in a group at 6 under par.
Phil Mickelson was one shot further back in a tie for 33rd at minus-5.