Woodland shot 2-under-69 on Saturday and moved to 11-under for the tournament. Rose shot three-under-68 and is 10-under at Pebble Beach.
Woodland has never won a major title. Rose won the U.S. Open in 2013.
Woodland had par saves on holes No. 12 and 14 to keep him bogey-free on the back nine.
Two-time defending champion Brooks Koepka is tied for third place at seven-under with Chez Ravie and Louis Oosthuizen. Rory McIlroy dropped to sixth place and sits at six-under. Chesson Hadley and Matt Kuchar are tied for seventh at five-under.
Aaron Wise began Saturday at five-under but shot plus-eight Saturday. He is tied for 48th place.