Harman, seeking his first major title, dropped two shots in the first four holes before birdies at the fifth, ninth, 12th and 13th took him to 12 under par, five ahead of American Cameron Young and six clear of Rahm.
The world number 26, showed great courage and composure after his shaky start, particularly as it happened just after Rahm had completed his round to ramp up the pressure.
"The start was tough, I hit a couple of loose shots," Harman told a news conference.
"It would have been easy to let the wheels start spinning, but I just doubled down and stuck to my routine. I knew I was still playing well despite those couple of bogeys."
World number three Rahm conjured up eight birdies in a majestic display of ball-striking to give himself a chance to win his third major crown and second of the year.
"That's the best round I've played on a links golf course ever," Rahm told reporters.
"Honestly, I'm just going to enjoy the afternoon with my family. There's nothing to be done. I've done a lot of good work this week to give myself an opportunity."
SIX BIRDIES
Young, runner-up at St Andrew's last year, quietly plotted his way around the course to pick up six birdies in a round of 66.
"My iron play has been great," he told reporters. "I've just hit a lot of greens and given myself plenty of chances. Honestly, I've putted really nicely too."
Norway's Viktor Hovland shot 66 to move to five under, level with Frenchman Antoine Rozner, Austrian Sepp Straka, Australian Jason Day and Tommy Fleetwood whose hopes of becoming the first English winner of the Open for 31 years faded with a 71.
"It was frustrating and obviously not the day we really wanted," Fleetwood said. "I played some good stuff but it wasn't enough today."
World number two Rory McIlroy collected three early birdies before losing momentum en route to a 69 and at three under par his hopes of claiming a fifth major title at the scene of his 2014 Open triumph are hanging by a thread.
Indian Shubhankar Sharma compiled a tidy 70 to finish at four under, level with England's Alex Fitzpatrick who carded 65 to finish two shots ahead of his brother Matt, the 2022 US Open champion.
Defending champion Cameron Smith of Australia made a battling 68 to end on one under, but world number one Scottie Scheffler needed two late birdies to complete a 72 and finish at four over.
American Scheffler admitted his chances of claiming a second major crown were slim.
"Unless a hurricane happens this afternoon," he said. "A hurricane and then some I think is what it's going to take for me to win."
