"It was hard to the very end," Simpson told PGA Tour Radio. "But I am very happy to get in there with a good score to start out."
Both he and Noren finished with five birdies and a bogey to lead a pack of five by one stroke.
"I played quite steady my front nine, which was the back nine, then I just scrambled my way around the front nine," said Noren, a nine-times European Tour winner who lost a playoff at the Farmers Insurance Open last month.
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"This was by far my best putting and chipping round of the season."
His only bogey came on his ninth hole, the par-five 18th.
PGA champion Justin Thomas, fellow Americans Morgan Hoffman and Daniel Berger, Canadian Mackenzie Hughes and South Africa Louis Oosthuizen, were all on 67.
Hughes had six birdies in his first seven holes before cooling off with bogeys at nine, 14 and 18.
Woods also had a solid start, standing two under after four holes before a bogey at his seventh hole and a double-bogey at his 12th proved costly.
Still the former world number one said he was pleased with his round of 70, which left him in a tie for 21st.
"Today was not easy" said the 14-times major champion who is making his third start of the PGA Tour season after a long absence following back surgery.
"It was tough all around today. One, the wind was blowing pretty hard. Then it was tough to make putts out there."
Yet Woods said he was happy with his putting on bouncy greens.
"I made a lot of key short putts for par," he said of his first back-to-back weeks of playing in three years.
He failed to make the cut last week at the Genesis Open in California.
Defending champion Rickie Fowler was one over par after three bogeys left him at 71 while Rory McIlroy put up a double bogey seven on the last for a two-over 72.
(Reporting by Gene Cherry in Salvo, North Carolina, editing by Greg Stutchbury)
