Djokovic wrote 'Kosovo is the heart of Serbia' on a camera lens on Monday, the same day when 30 NATO peacekeeping troops were hurt in clashes with Serb protesters in the Kosovo town of Zvecan
Spain's Rahm, whose first win on the US PGA Tour came at this event two years ago, is one stroke ahead of world number one Justin Rose and American Doug Ghim, who both shot nine-under 63s.
"There's a lot of happiness going on in this area for me, so I just feel I'm going to play good when I tee up," Rahm said.
Fourteen time major winner Woods birdied his last hole of the day to post a 70 and is eight shots back of Rahm at the Torrey Pines Golf Course in La Jolla, California.
Taiwan's C.T. Pan is fourth after shooting a 64 and Jordan Spieth is alone in fifth with a seven-under 65.
The 24-year-old Rahm came into the event as one of the hottest players on the tour and his start reflected that as he eagled both par-fives on his front nine, including the opening hole.
Rahm, who began on the 10th tee on the North Course, sailed his approach to inside of 10 feet from the flag and then rolled in the putt for an eagle.
The world number seven also eagled the par-five 17th. He finished with seven birdies and one bogey which came on his back nine.
Rahm's last win came in the Hero World Challenge in the Bahamas in December. He has four straight top-10 finishes.
He is the youngest champion in tournament history, winning at age 22 and two months.
"This is probably my favourite area," Rahm said. "If you take into account the town and the vibe around Torrey Pines, this is my favourite golf course and golf tournament."
Rose, who won the Turkish Airlines Open on the European Tour in November, is coming off a tie for 34th at last week's Desert Classic near Palm Springs.
Rose shot a bogey-free 63 that included an eagle and seven birdies. Like Rahm he began with an opening eagle on the par-five 10th.
Woods, who is playing his first official tournament since winning last year's Tour Championship, finished with five birdies, three bogeys and 10 pars for a share of 53rd place. He played on the South Course which is the more difficult of the two.
"A couple under par on the South Course is not something to sneeze at, but now I have to shoot a low one tomorrow," said Woods, who hit half his fairways and made one 29 foot putt on the 13th hole.
"I felt pretty comfortable with everything today," Woods said. "I felt like I drove it halfway decent and irons were good.
"Playing at competitive speed again, I didn't quite hit all my irons pin high like I normally do. That's something hopefully I'll have a better handle on tomorrow.
"It was nice to have some juice flow in the system again, it's been a while."
He has won this event seven times, but not since 2013. Woods also captured the last of his 14 majors at the US Open on this course in 2008.
World number eight Rory McIlroy fired a one-under 71 in his Torrey Pines debut and defending champion Jason Day of Australia shot five-under 67 and is tied for 15th.
Djokovic wrote 'Kosovo is the heart of Serbia' on a camera lens on Monday, the same day when 30 NATO peacekeeping troops were hurt in clashes with Serb protesters in the Kosovo town of Zvecan
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