Woods, the top-ranked player on the PGA Tour, struggled with his play Saturday. In rainy weather conditions, the second-round leader shot 2-over 72 and Harrington caught him, despite a 1-over 71.
Two of the world's best golfers have overall scores of 6-under-par 204.
"I know he's got a very good record when he's leading going into the final round," Harrington said about Sunday's closing round. Woods' record when leading after three rounds is 38-3.
Japan's Shingo Katayama (71) is alone in third place at 4-under 206.
Takuya Taniguchi (68), also from Japan, is alone in fourth at 3-under 207; while Keiichiro Fukabori (69) of Japan and England's Justin Rose (72) share fifth place at 2-under 208.
Rose lost a chance to join the leaders after a late triple bogey.