While Rose eagled two of the four par-fives on the easier North layout to move one ahead of the field with a sparkling seven-under 65, Woods dropped six shots in six holes from the 12th to end the day 11 strokes off the pace.
It was the 41-year-old's highest opening round to begin a PGA Tour season since he turned professional in 1996.
"I was one under through 11 and I was in a good spot to really shoot a good round ... but it went the other way," Woods told Golf Channel. "I hit some bad tee shots, made a bad three-putt there on 13 ... just kept compounding problems and mistakes out there."
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Watched by huge galleries, Woods made a faltering start with an opening bogey after setting off in the company of world number one Jason Day and third-ranked Dustin Johnson.
After finding the right rough off the tee at the par-four first, he missed the green to the right with his approach and failed to get up and down in two.
Though he flared several iron shots to the right on the outward nine, Woods used his renowned scrambling abilities to save pars from several unpromising positions to reach the turn in one-over 37.
He gave fans something to celebrate with birdies at the 10th and 11th but his round unravelled over the next four holes. Wayward tee shots led to bogeys at the 12th, 13th and 14th and Woods ran up a double at the 15th after hooking his drive into a hazard.
He missed a five-foot putt to bogey the par-four 17th before signing off with a nine-footer to birdie the last.
Woods, who was sidelined by back pain and a series of surgeries, had not competed in a regular PGA Tour event since August 2015 and is also adjusting to his new TaylorMade clubs.
(Reporting by Mark Lamport-Stokes in St. Augustine, Florida; Editing by Frank Pingue)
