Noren, a nine-times European Tour winner, birdied his final hole for a one-shot lead over PGA champion Justin Thomas, Morgan Hoffman and Daniel Berger on what was a windy day at PGA National in Palm Beach Gardens, Florida.
The Swede had five birdies and a bogey for the good start.
“I played quite steady my front nine, which was the back nine, then I just scrambled my way around the front nine,” Noren told PGA Tour Radio.
“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.
Woods was 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.
“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, eight bogeys in his second round ending his play.