He tied for 58th in the Masters after becoming the youngest player to make the cut at Augusta National, and said he will play in a US Open qualifier in two weeks in Dallas.
Ernie Els and Rickie Fowler head a group of 10 players at 5 under.
Defending champion Jason Dufner shot 70, and 2011 winner Bubba Watson had a 73.
Eighty-four players were at 1 under or better on the rain-softened TPC tract, with 30 players within 4 shots of the lead. Heavy rains that spawned at least two tornadoes in the New Orleans area on Thursday left TPC's normally firm greens soft.
Players were allowed to lift clean and place their golf balls in the fairways, which especially benefited the later players after an early mist gave way to sunny and ideal scoring conditions.
Barnes missed the cuts in seven of his last eight events. He jump-started his round with the eagle on the par-5 second hole, hitting a 3-wood from 290m to 2.7m.
"That kind of got the round going," he said. "I was 3 under [after a birdie at the fourth] and those are the kind of starts I need."
Barnes birdied the par-5 11th, had four consecutive birdies on No 13-16 and finished with a birdie on the par-5 18th. He hit all 14 fairways in regulation and had 26 putts.
Barnes said a new, albeit unusual, pre-putt routine has helped his focus. He uses a left-hand low grip for the practice stroke and a "claw" grip for the actual stroke.
"If you watch a lot of guys that [putt] cross-hand, their shoulders are able to stay very level," he said. "It's a good way of getting the feeling of hitting down on the ball."
The stage might be set for another Barnes-Glover shootout, similar to the 2009 US Open at Bethpage in New York when the duo battled in the final three rounds. Glover eventually overtook Barnes in the final round to win his only major championship.
Glover birdied his first four holes, playing the back nine first, to get to 6 under. After a bogey on the sixth, he closed the round with birdies on the eighth and ninth holes.
AP
