After a par on the sixth, the 2019 Open champion drained a 24-foot putt for a birdie two on the next and closed out his front nine with two more birdies to turn in six-under 30.
Lowry, who has not won an individual title in four years, scrambled for a par five at 10 when he hacked out of long rough before holing a 10-footer.
Birdies at 11 and 12 ignited hopes of a 59 at the par-70 venue before a sloppy bogey on 13 when he three-putted from 18 feet.
But the world number 53, who attracted large galleries around the spectacular links course, rolled in a 19-footer for birdie at 15 to return to -11 before finishing with three pars.
"I thought about it," Lowry, who has not returned a top-20 finish since February, said when asked if a sub-60 round entered his mind.
"Yeah, I did think about it until I three-putted 13. No, I did have a little laugh. Out in 30 and the back nine is only 34, so, yeah, you do kind of think about it a little bit. "
He added: "I know I made a little mistake on 13, but I think I did a good job after that of playing the rest of the round really well and giving myself a couple of chances on the way in. "
