The 25-year-old, who needs to win this week to make his debut in the season's opening major at Augusta National, birdied his last two holes for a three-under-par 69.
That left him at nine-under 135, a stroke in front of compatriots Johnson Wagner (67) and defending champion Anthony Kim, who rocketed up the leaderboard with a best-of-the-day 64 at Redstone Golf Club.
Triple major winner Padraig Harrington of Ireland (69) was a further shot back at seven under, level with Americans Josh Teater (72) and first-round leader Jimmy Walker (74).
Kirk, who won twice last year on the PGA Tour's satellite circuit, the Nationwide Tour, was thrilled with the way he bounced back from a double-bogey at the par-five fourth, his 13th hole of the day, where he three-putted.
"I just made a real mess of that hole after making two birdies (at the second and third)," the former University of Georgia standout told reporters. "So to get both of those shots back on eight and nine was really great."
Kirk played Augusta National four times as a student, through invitations by University of Georgia alumni, and he would dearly love to play his way into the Masters.
"That's a huge goal of mine," he said. "That's the one tournament that I grew up watching and that I would love to play in. I would love to win that tournament one day." Kim, who missed much of last season due to a thumb injury, was delighted to regain a glimpse of the form that has helped him win three PGA Tour titles.
"I'm really pleased with my round today, obviously because of the score but also because my coach and I have been working so hard on my game," the 25-year-old said.
"I've been playing out of trees and bushes (in recent months), and that's pretty tough. But today I got off to a pretty solid start.
"It's been a tough stretch, but I'm glad to have this one under my belt. Now I can draw back on this and know that when I do get it going, I can shoot under par." Harrington, who has not shaved this week after forgetting to bring a razor with him to Texas, was also happy to be in contention for the title going into the weekend.
"It's always where you want to be," the 39-year-old Dubliner said. "I feel like I've played really comfortable to be where I am." Among the other big names, British world number two Lee Westwood returned a 72 to lie five shots off the pace, level with Masters champion Phil Mickelson who carded a 70 while being watched by former President George H.W. Bush.
The cut fell at even-par 144 with former major winners Angel Cabrera of Argentina, Americans Rich Beem and Justin Leonard, plus South Korean Yang Yong-eun, among those failing to advance.
