While Watson launched his title defence with a battling one-under-par 71 in blustery conditions at the TPC Louisiana, journeyman Duke capitalised on a strategy of patience to card a flawless 65.
Bidding for his first victory on the US PGA Tour where he has been a runner-up three times, the 43-year-old Duke rattled up seven birdies to finish level with compatriot Cameron Tringale.
"You have to be patient out here," Duke said after ending his round with birdies at the 16th and 18th. "You can't hit it (to) four, five feet on every hole because there are so many mounds on the greens.
"You have to take your 30-foot putt here, 30-foot putt there and take what it gives you. If you roll one in, you roll one in. I did that a couple of times today."
Americans Steve Stricker and Chris Stroud, Swede Daniel Chopra and three-times major winner Ernie Els opened with 66s while Ben Curtis, who won the Texas Open on Sunday, was among a small group a further stroke back.
It was a tough day, though, for defending champion Watson who was competing in his first event since winning the opening major of the year at Augusta National 18 days ago.
"I knew I needed to be here being defending champ ... but mentally I'm not here," said the long-hitting left-hander who played in a high-profile grouping with world number seven Stricker and fellow American Webb Simpson.
"It's overwhelming what happens when you win a big event like that (the Masters). The mental grind of making sure you treat everybody the same, making sure you sign everybody's autograph ... it's just time-consuming.
"I'm so out of it right now, golf is a hard thing to do right now. I just haven't had time to rest yet and take a deep breath. I'm just exhausted. Hitting the ball full bore tires me out."
Watson, who beat South African Louis Oosthuizen at the second extra hole to win his first major title at the Masters, said his main aim this week was simply to make the cut.
"That's what me and my caddie joked about all day," Watson smiled. "He said a good week is going to be making the cut, then a great week is going to be anything in the Top 25.
"That's what I kept grinding about and somehow I had a few birdies."
Watson won last year's New Orleans Classic in a playoff with Simpson.
British world number two Luke Donald made an uncharacteristically erratic start with a 73 that included four birdies and one double-bogey while 10th-ranked fellow Englishman Justin Rose opened with a 72.