American Spieth, one of the most exciting young players in the game who was the PGA Tour's Rookie of the Year in 2013, fired a nine-under-par 63 on the easier North Course at Torrey Pines, one of two venues hosting the event.
Helped by a slight adjustment to his putting technique since the opening round, Spieth covered his back nine in five-under 31 in calm, mainly overcast conditions to post a 10-under total of 134.
That left him a stroke in front of overnight leader Stewart Cink, who birdied his final hole, the par-five ninth, for a 71 on the brutally difficult South Course. Long-hitting Belgian Nicolas Colsaerts was a further shot back, after a 67.
World number one Woods, seeking an eighth victory at this tournament and a ninth overall at Torrey Pines, failed to birdie a single par-five for the second day in a row as he carded a 71 to finish a distant nine strokes off the pace.
"I was a yard off, a foot off all day," a frustrated Woods, who declined an interview with Golf Channel immediately after signing his scorecard, later told reporters.
"(It was) one of those days where I just could not get anything going because of it, and then when I did have an opportunity to get something going, I stubbed my foot by three-putting twice out there."
While Woods mainly struggled on the North layout, Spieth took advantage with a nine-birdie display, largely because of his improved putting.
"I changed it a little bit, put my weight a little more on my front foot today," said the 20-year-old Spieth, who clinched his maiden PGA Tour title at last year's John Deere Classic during a sensational debut season.
"I had noticed that was kind of causing a little bit of a tendency to push it on the shorter putts for me. So I was more stable with the ball and I plotted my way around an easier golf course in some nice conditions."
Asked how he had managed to cope with the huge crowds and heightened media focus while playing in a group that included 14-times major champion Woods, Spieth replied: "I've spent a lot of time with him.
"It wasn't the first time I had teed it up next to the guy I've watched growing up. He is easy to play with, he talks to us. Similar to Jimmy (Walker, the third member of the group), they are both very easy, fun guys to play with.
"We played with some good speed, kept the tempo up and I just played my game ... didn't even see a scoreboard and really didn't know where I was at until I got to number nine. That's the way you want it to be out there when you're in the zone."
San Diego native Phil Mickelson, a three-times champion at Torrey Pines, carded a 73 on the South Course to finish at two under but was forced to withdraw from the event because of muscle pain in his back for a second consecutive day.
"I am a little bit torn right now on the best plan of action," the American left-hander told Golf Channel after bogeying his final hole, the par-five 18th, despite having a sand wedge approach from the middle of the fairway.
"I love this tournament and I want to play here. I love playing in San Diego, and this is a place where I grew up playing.
"And part of me also says I am getting into bad habits and if I hit it in this rough here, I've really got an issue because it's awfully hard to get it out and it jars it (his back)."
His management team later issued a statement confirming Mickelson's withdrawal from an event he last won in 2001.
The cut fell at level-par 144 with Fiji's former world number one Vijay Singh and 2012 champion Brandt Snedeker among those failing to advance.
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