The 29-year-old Brown, a one-time European Tour winner who came through qualifying to earn his place in the year's final major, took advantage of benign late conditions.
He sank an eight-foot birdie putt at the last to move top of the leaderboard with a bogey-free round.
"It did get dark out there and got a little bit tricky but the wind dropped as well," Brown told reporters.
"You've got those bright yellow scoreboards out there so it's hard to miss, but I felt comfortable with my game."
Lowry, the 2019 Open champion, birdied the last hole to complete a flawless five-under-par 66 and move two shots clear of American Justin Thomas, who took the early clubhouse lead at the 152nd Open with a 68 at a blustery Royal Troon.
"Actually, I holed a putt out there today and I thought that's the first time I've led the Open for five years," Lowry told Sky Sports.
"I love this tournament and I'd dearly love to give myself a chance to win it again. If I can keep holing putts I can be dangerous."
American Xander Schauffele, Swede Alex Noren, England's Justin Rose and Joe Dean, American Russell Henley, Denmark's Nicolai Hojgaard and Canadian Mackenzie Hughes finished on two under with world number one Scottie Scheffler among a group a further shot back.
Several big names struggled, however, in testing conditions on the Ayrshire coast, not least US Open champion Bryson DeChambeau who was six over after 11 holes before making eagle on the 16th to finish at five over.
POOR START
World number two Rory McIlroy also endured a poor start to his bid for a fifth major title after carding a miserable 78.
Thomas, who has never had a top-10 finish at the Open and missed the cut last year, birdied the second, fourth and seventh as well as the Postage Stamp par-three eighth.
A double-bogey six at the 12th and a bogey at 13 stalled his progress but he holed a 17-foot par putt at the 15th and birdied the last two holes.
"I played so well today but gave a couple back at the beginning of the back nine," former world number one Thomas said.
"I'm playing really well and feeling confident about my game. It's nice to get off to a good start in a major."
Northern Ireland's McIlroy bogeyed the first and dropped two shots at the Postage Stamp as his bunker escape rolled back into the cavernous sand trap.
Things got worse when he bogeyed the 10th and he emerged with a double bogey at the 11th after hitting his tee shot out of bounds over the railway track skirting the course.
Defending champion Brian Harman uncharacteristically endured an up-and-down front nine with two birdies and three bogeys before finishing on two over par.
Large galleries headed for home favourite Robert MacIntyre who was joined by Spain's Jon Rahm and England's Tommy Fleetwood.
Last week's Scottish Open winner MacIntyre started solidly but undid an early birdie with a missed par putt at the fifth and finished on one over, one shot ahead of Rahm, as Fleetwood struggled to a round of 76.
