Herron, who has four PGA TOUR wins, finished with a 65 and was at 12-under 198 for the tournament. Clark (66) and second-round leader Joe Ogilvie (69) were tied for second one stroke back. Kenny Perry (67) was at 9-under, and Jeff Brehaut (65) and Charlie Wi (66) were four strokes behind.
Herron and Clark, who will be paired together in Sunday's final round, slogged through the front nine before Herron bogeyed No. 10 to fall to 9-under.
On the 15th hole, Herron made a 4-foot birdie putt and Clark added one from 3 feet. The next hole, Herron sank a 14-footer and Clark answered with another from 3 feet.
It stayed that way until the par-5 18th, when Herron got up and down from the sand to make birdie and move to 12-under.
Clark couldn't match it, missing a 6-footer.
Clark, who finished second last week at the John Deere Classic and second in the 2006 Masters, has never won a PGA TOUR event. He's been suffering with a bulging disc in his neck, but took a cortisone shot three weeks ago in an effort to make the Presidents Cup team and reach the playoffs in the FedExCup competition.
Ogilvie, who came into Saturday's play leading by two strokes, went 0-for-11 on putts in the 15 to 20 foot range, including misses for birdies at Nos. 2-6. After playing the first 42 holes with only one bogey, he carded one No. 7 and another on the eighth.
On No. 8, Ogilvie drove the ball just off the fairway, but it hit an object somewhere just in the rough and kicked about 20 yards into trouble, forcing him to hit a save shot back to the fairway to continue.
Ogilvie rebounded with an eagle-birdie combination on the 15th and 16th holes to move back to 11-under, but had to settle for par despite a good opportunity on No. 18.
Perry, a nine-time TOUR winner and the 2003 champion at this tournament, hasn't had a victory since 2005 after having right knee surgery last year to fix a torn ligament.
The rehab nearly took Perry to the brink of retirement because he said he had to learn how to make his right leg flex around his swing again.
But he regained his form with a 63 in the final round of the Memorial last month to tie for third, and has reeled off three more finishes of 15th or better heading into this tournament. Perry has played all three rounds under par this week.
Wi birdied the final hole to join Brehaut at 8-under. Brehaut had two eagles on Saturday and four for the tournament. The five-time participant in qualifying school, Brehaut has only made three cuts in 11 events so far this year and last won a Nationwide Tour event in 1997.