Fowler, who shared the overnight lead with Bryson DeChambeau rolled a nine-foot birdie putt into the heart of the cup at the 18th hole of TPC Scottsdale to move atop the jammed leaderboard, firing a four-under 67 for 14-under 199.
World number two Rahm, a fan favourite who played college golf in Arizona, shook off a disappointing finish to his second round with a six-under 65 that included five birdies from the 11th.
Jon Rahm is quickly taking the PGA Tour by storm, and he continued his success Thursday at the Waste Management Phoenix Open.
Five-time major victor Phil Mickelson carded a 66 to head a group on 201.
Two years ago, Matsuyama beat Fowler in a playoff after the American plonked his tee shot into a water hazard at the fourth extra hole. "Just a little off in certain areas and just kind of tried to piece things together".
Rahm, seeking a fifth worldwide title in the past year and his second USA tour victory in three weeks, was two-under through nine and responded to a bogey at the 10th with birdies at 11, 13, 15 and 16.
"Every hole, all these fans and I'm guessing tomorrow afternoon there's going to be a lot more people especially in my group playing with Phil again". He tied for 12th last week at Torrey Pines in San Diego.
"The first 62 I shot was in Manila two years back on the Asian Tour, " he said. "You need to play really good golf", Rahm said.
However, Hideki Matsuyama's hopes of winning the tournament for a third consecutive year evaporated when the Japanese world number five pulled out before hitting a shot, citing a left thumb and wrist injury.
Justin Thomas went the other way, wasting a brilliant opening run with another late meltdown.
But then had to fight to shoot an even-par 71 after a back-nine meltdown. "That pretty much sums it up".