Furyk, who opened with a dazzling seven-under-par 63 in the elite World Golf Championships (WGC) event, followed up with a five-birdie 66 on another steamy day at Firestone Country Club.
The veteran American scrambled superbly to cover his back nine in a flawless two-under 33, posting an 11-under total of 129 to finish two strokes in front of Spaniard Rafael Cabrera Bello, who fired a 65.
South African Louis Oosthuizen, the 2010 Open champion, was at eight under, also after a 65, with laid-back American Jason Dufner a further stroke back after returning a 66.
Woods, who had totalled 33 putts on the slick Firestone greens in the opening round, mixed four bogeys with two birdies for a 72 to end the day a distant 13 strokes off the pace.
Furyk, who clinched the most recent of his 16 US PGA Tour victories at the 2010 season-ending Tour Championship, was delighted to retain control of the tournament after starting the day two ahead of the chasing pack.
"At times you go out there and fire a low score one day and it's hard to follow it up the next and keep that momentum going," the smiling 42-year-old said.
"So I was able to kind of keep that momentum. I knocked in a couple of good putts today and made some great saves actually, some good par saves, to keep the round going."
Furyk saved par four times from bunkers on his back nine before ending the day with his lowest 36-hole score on the US PGA Tour.
"I thought it was key to get off to a nice start and see some putts go in," he said. "I made a good birdie putt at two and saw some birdies go in on the front nine, good putt at nine, and kind of off to the races.
"The back nine was a little bit more making some good saves. I slipped a couple of birdies in there along the way, as well. Good round."
While Furyk was in an upbeat mood, Woods was a frustrated figure after needing 29 putts in the second round.
"I hit it good, made nothing (on the greens)," the 14-times Major winner said. "I figured something out finally on the 17th hole, but too little, too late.
"I get in these little spells where it's hot or cold. Generally I was a decent putter over the years, but lately it's been very streaky, I'm making everything or I make nothing."
Woods, who has triumphed a season-high three times on the 2012 US PGA Tour, explained that his putting woes stemmed from the wayward path of his putter.
"I had my lines good, but it's just setting my path out," he said. "I was trying to marry the two. I was trying to figure it out last night on the putting green.
"Yesterday was bad putts. Today I had good speed and just still not quite right. And the putts I did (hit) pure, they were just lipping out. So that's fine."
British world number one Luke Donald, struggling on the greens in the late afternoon, carded a 69 to end the day six strokes behind Furyk while defending champion Adam Scott of Australia posted a 70 for a one-over total.
Former US Open winners Rory McIlroy and Graeme McDowell, both of Northern Ireland, fired matching 67s to sit eight shots off the pace but Open champion Ernie Els returned a second successive 73 to languish joint 62nd in the 77-man field.
"I played solid today," said the mop-haired McIlroy, who romped to victory by a staggering eight shots at the 2011 US Open. "I (double-bogeyed) the first, but apart from that, it was pretty good.
"Making six birdies, it was nice. I'm just trying to keep improving on my score. If I can get lower for the next two days, I won't be far off."