Woodland finished 54 holes at 8-under 208 on a day that only 18 players broke par despite ideal conditions on the South Course at Torrey Pines.
Overnight leader Jordan Spieth managed a 3-over 75 to fall into a share of second place at 7-under with Marc Leishman (even-par 72).
Morgan Hoffmann and Pat Perez each shot 72 to finish the day tied for fourth at 6-under 210.
The low round of the day went to Argentine Andres Romero, who fired a 5-under 67 to grab a share of sixth at 5-under with Rory Sabbatini (69), Brad Fritsch (72), Nicolas Colsaerts (75), Will MacKenzie (70), Scott Stallings (72) and Ryo Ishikawa (69).
Seven-time winner Tiger Woods missed the secondary cut for the first time in his career with a 7-over 79, leaving him at 6-over 222.