Vaughn Taylor, who bogeyed the 18th hole, stands alone in second place at 8-under-par 63. Greg Chalmers and Daniel Chopra share third place a 7-under-par 64, while Ted Purdy and Jeff Brehaut are one stroke further back at minus-6.
Coceres opened his round at the par-5 10th and dropped his third shot within 9 feet of the cup to set up a birdie. At the par-3 12th, he stuck a 5-iron 3 feet from the cup for birdie No. 2.
The Argentine sank his third birdie at the par-5 14th from 9 feet out. Coceres really caught fire from there.
Coceres dropped an 8-iron three feet from the cup to set up birdie at the par-3 16th. He came right back with a 12-foot birdie putt at the 17th. Coceres closed his opening nine by holing out from 45 feet for his sixth birdie.
'It's very important to have a lot of confidence. I wasn't really thinking about it,' said Coceres of his chances of shooting 59. 'I got some birdies, and got to four-under, then that gave me more confidence to attack the later holes.'
The 40-year-old was not done either. He rolled in a 15-foot birdie putt at the first and came right back with a two-putt birdie at the par-5 second to cap a run of five consecutive birdies.
Coceres cooled off to card five pars in a row. He then rolled home a 36-foot birdie try on the eighth at TPC at Deere Run.
'I felt that the front nine was easier than the back nine, so that's what I probably could do better,' said Coceres, who broke his arm playing soccer and missed most of the 2002 season while recovering from that injury.
Taylor had a chance to match the course record of 61, but he followed both of his eagles with bogies. The 28-year-old opened with an eagle at the par-5 second, but stumbled to a bogey three holes later.
The tour rookie came right back with a birdie at the par-4 sixth. Taylor made it two straight with a birdie at No. 7. He then capped a run of three straight birdies at the eighth to move to 4 under.
Around the turn, Taylor moved to 3 under on the par-5s alone as he birdied the par-5 10th. He climbed to 6 under with a birdie at the par-4 13th.
Taylor picked up his sixth birdie at the par-3 16th. He jumped into a share of the lead at minus-9 with an eagle at the par-5 17th. However, he stumbled to a bogey at the last to fall one shot behind Coceres.
'I felt like I made a good stroke there, it just didn't go in,' said Taylor of his putt at the last. 'I didn't feel any different. I felt fine.'
Fifty-year-old Jay Haas posted a 4-under-par 67. He shares 11th place with Jason Bohn, Jonathan Byrd, Robert Gamez, Todd Hamilton, Chris Smith, Kevin Sutherland, Jim McGovern and 1999 champion J.L. Lewis.
Defending champion Vijay Singh could not get going during his round. He spread two birdies and two bogeys over his opening nine before a pair of birdies on his second nine got him to 2-under-par 69, where he shares 41st.