American Furyk, whose only major victory came in the 2003 US Open at Olympia Fields, bogeyed the treacherous par-four sixth but safely parred his other eight holes to reach the turn in one-over 35.
That left him at level par overall for the year's second major, a stroke in front of compatriot Webb Simpson, who had charged into contention with four birdies in five holes from the seventh.
Little known American Michael Thompson was a further shot back at two over in a tie for third after signing off with a three-under-par 67 to grab the clubhouse lead on a foggy afternoon at the Olympic Club.
Level with Thompson, who had led after the opening round, were Northern Ireland's Graeme McDowell and American John Peterson.
McDowell, US Open champion at Pebble Beach two years ago and joint pacesetter overnight with Furyk, had completed nine holes while Peterson had five remaining.
On a gripping afternoon at Olympic where most of the field struggled over the brutal first six holes, Furyk was handed the outright lead at the par-three third when McDowell bogeyed.
McDowell's tee shot ended up just short of the green from where he hit a poor putt eight feet past the cup and failed to sink the par putt coming back.
Though Furyk squandered a birdie chance at the fourth where he struck an exquisite approach to seven feet, he moved two shots clear after McDowell bogeyed the par-four fifth.
Furyk made his first mistake of the day when he bogeyed the par-four sixth after hitting a poor chip from behind the green but remained two ahead after McDowell also bogeyed the hole.
Though Furyk parred the next three holes, his lead was cut to one by the charging Simpson.
Three-time champion Woods, who was five strokes off the pace overnight, suffered a nightmare start, going bogey, bogey, double-bogey over the first three holes.
He found the right rough off the tee at the par-four first, dumped his approach into a greenside bunker at the second and took three shots to reach the green at the par-three third.
After parring the fourth, Woods then bogeyed the fifth and sixth to drop to 10 over par in his bid to end a major title drought of four years, a distant 11 strokes off the lead.
Though Woods rallied with birdies at the eighth and 14th, he languished in a tie for 28th place.
