Spain's Rahm, ranked third in the world, fired a six-under par 66 that featured an eagle and five birdies on the Torrey Pines South Course to move within two strokes of Ryder's lead.
Ryder, who started the day with a three-shot cushion over Brendan Steele, needed all of his patience on the way to an even par 72 that included an early bogey offset by one birdie and left him on 12-under par 204.
Rahm, who started the day eight adrift, said his goal was to "catch up as much as possible."
After following a birdie at the fourth with a bogey at the fifth, Rahm exploded to pick up five strokes in the space of four holes with birdies at the sixth, seventh and eighth and an eagle at the ninth, where he rolled in a 12-foot putt.
"I just kept giving myself pretty much the easiest putts," Rahm said. "Then on nine, put it in the fairway and was contender for best shot of the year for me already, which was the second shot into the green. I tattooed that five-wood and ended up having 10 feet for eagle."
Rahm added a 13-foot birdie at the 15th, but had to battle to save par at the last two holes after finding fairway bunkers at both.
Even with some disappointing shots, Rahm said, he "never really lost composure."
"I stayed really patient out there and took advantage of the holes I could take advantage of and played what I would say was very smart throughout the round," said the Spaniard, who won the first of his nine US PGA Tour titles at Torrey Pines in 2017 and captured his first major on the course in La Jolla, California, at the US Open in 2021.
Number one on the line
Rahm, who opened the year with a victory at the Tournament of Champions in Hawaii and won the American Express in La Quinta, California, last week, is seeking to become the first player since Dustin Johnson in 2017 to win three straight PGA Tour starts.
With a victory he could overtake Northern Ireland's Rory McIlroy for number one in the world. Rahm, who has spent a total of 43 weeks at No. 1, most recently from July 18, 2021 to March 26, 2022, could even regain the top ranking with a lower finish, depending on where McIlroy finishes in the Dubai Desert Classic.
McIlroy was one off the lead after the first round ended in the weather-hit tournament in Dubai, which is now scheduled to end on Monday.
Ryder, meanwhile, is chasing a first US PGA Tour title in his 147th career start. He saw a bevy of birdie chances go begging as he parred the last 12 holes, unable to get a five-foot birdie putt to drop at the 13th and watching an eight-foot birdie attempt burn the edge at 14.
He had one more look at the 18th, but his 10-footer didn't drop.
American Tony Finau thrust himself into contention with an eight-under par 64. Finau teed off on 10 and had four birdies in his first nine holes before holing out for eagle from the fairway at the first. He added birdies at the sixth and ninth for solo third on 208.
