Rose led by as many as six strokes, but his second double bogey of the day, at the par-four 14th, trimmed the lead and he closed with a bogey at the par-five 18th, where his second shot from the left rough found the water.
Defending champion Jason Day of Australia will need a huge rally to repeat.
Scott, making his debut at this tournament, didn't make a birdie until the ninth hole and missed a 20-inch par putt on the front nine.
Scott was the only player to give Rose a serious run, and it didn't feel like much. When I messed up I messed up big, but I'm pleased with the way I responded to the doubles. His work with the upgraded version of his TaylorMade P7TW irons was substandard and led him to admit that he felt "a little off" in that area, saying he wasn't "as clean as I like" through impact.
"Yeah, tee shot on 1 was nice because normally I haven't really played that hole all that well", said Woods, who was paired with Tony Finau and Xander Schauffele, who played at San Diego State.
Justin Rose tracked the flight of his 3-wood until he saw it splash into the pond guarding the 18th green at Torrey Pines.
"Probably harder than the heart surgery itself, watching this weekend", Rose said. McIlroy holed out with a lob wedge on the par-4 16th hole for his second eagle and shot a 65.
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"I feel like I knocked some rust off, which was great this week", Woods said.
"Sixty-five around the south course, I don't care what the conditions are, it's a good score". You have to go out and get it. "I love you, and we'll see you soon". "I'm still lagging way behind".
Third-placed Rahm warned not to expect a repeat of his charge to victory in 2017, when he played the final nine holes in 30. A couple under par on the South course is not something to sneeze at, but now I have to shoot a low one tomorrow.
"It's very hard to do that again".
The putts finally started to drop on the back nine.
Tiger Woods, an eight-time victor at Torrey Pines, picked up one shot in a round of 71 and finished at five under.
That was his 14th major title, a number he has not added to in the ensuing decade.