The 2012 Masters champion carded a three-under-par 68 to move 15 under overall, two clear of Kevin Stadler, with a trio of players a further shot back.
Joint overnight leader Matt Jones slipped four shots off the pace with a 72, the same as defending champion Phil Mickelson who finished the day three under.
Watson did not show the sort of form that brought him rounds of 64 and 66 on Thursday and Friday, struggling with driver in hand in particular, and found himself in the sand on several occasions.
But a series of impressive wedge shots on the back nine ensured he carded just one bogey, to go with four birdies, improving his position as he looks for his first title since that Masters win.
Stadler, meanwhile, was one over for his round through 12, but five birdies in the final six holes lifted him clear in second place.
A shot behind him were Harris English, Hideki Matsuyama and Ryan Moore, who put himself firmly in the mix with a seven-under round of 64, six of his seven birdies coming on the back nine.
Also surging into contention was Brendan Steele, whose blemish-free round of 62 put him 11 under alongside Hunter Mahan and Jones.
The round was reminiscent of Mickelson, who flirted with a 59 on his way to the title last year, but there were no such fireworks from the defending champion this time as he slipped back into a tie for 40th.
Scotland's Martin Laird remained the leading European on seven under despite a 71, with England's David Lynn and David Lingmerth of Sweden two shots further back.