Woodland, a three-time US PGA Tour winner in search of his first major title, stretched his two-shot overnight lead to as many as four on the front nine.
While he couldn't maintain that lead, a back-nine birdie at the 11th and three straight par saves at 12, 13 and 14 kept Rose at bay, the Englishman whittling the gap to one stroke with a birdie at the 18th to cap a three-under-par 68.
Woodland had an 11-under-par total of 202, Rose was at 10-under 203.
It was another three strokes back to a group headed by two-time defending champion Brooks Koepka, who stayed within striking distance of a fifth major title with a 68 for seven-under 206.
Koepka, who could become just the second player, and the first in more than a century, to win three straight US Opens, was joined at seven-under by fellow American Chez Reavie, whose three-under 68 put him in the top 10 going into the final round of a major for the first time, and former Open Champion Louis Oosthuizen of South Africa who carded a one-under 70.
Northern Ireland's Rory McIlroy, trying to add to his cache of four major titles for the first time since 2014, was alone in sixth after a one-under 70 for six-under 207.
He was a stroke in front of Americans Matt Kuchar and Chesson Hadley, who both posted 70s to land at five-under.
Tiger Woods, who began the day nine shots off the pace on even par, was unable to make headway, finishing with five birdies and five bogeys to remain even for the tournament.