Woods, who has not scored a tournament victory since August 2013, needed a birdie on the par-four final hole to finish level with eventual victor Paul Casey. From 185 yards, his 7-iron came up some 40 feet short, and his birdie putt to force a playoff was 2 feet short.
The Englishman finished at 10 under, while Woods and fellow American Patrick Reed ended at 9 under.
Woods and his massive following went dormant after an opening birdie to briefly share the lead.
There was nevertheless much for Woods to be positive about, as he registered his best finish since winning five events in 2013.
Former world number one Woods, playing his fourth US PGA Tour event since returning in January from spinal fusion surgery last April, fired a three-under-par 68 for a four-under total of 138.
"It feels good to be up here", Woods, who has not won since 2013, said.
"I was walking down the sixth and looked over to the fourth green", Rose said. "It was one of those days I kept getting half-clubs".
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As a massive gallery - energized by Woods's mere presence in a tournament he had never played, then whipped to the point of frenzy by his late Sunday charge - pressed in around the green, Woods studied his long birdie putt from all angles, took his putter back, and unforgivably left it almost three feet short.
Thousands lined the fairway at every hole Woods played and his galleries dwarfed those for the rest of the field with attendance at a tournament that past year drew 112,000 soaring to more than 150,000 for his first appearance.
He's expected to tee it up at the Arnold Palmer Invitational next week in his final Masters tune-up. I hadn't played there in a couple years because of my back and I wanted to play there a couple years ago.
On his way off the course, Woods stopped to sign autographs, showing a good deal more patience than he had during his heyday."Wow, what an unbelievable week. people, atmosphere, adrenaline, back nine on Sunday, man I've missed this", Woods later tweeted. He got up-and-down from a bunker short of the green at No. 11 for birdie, stuffed his approach to a foot on No. 12 and then holed a 20-foot birdie from just off the green at No. 13.
"It was a very good week", the 42-year-old told Sky Sports.
He led compatriot Justin Rose, who picked up one stroke in eight holes to improve to nine-under. Our paths crossed on the golf course quite a few times [Saturday] because there are a couple holes the way the course is laid out and intersects where our groups were waiting for one another or vice versa. I played some really good golf.
He dropped his only shot on the next hole when his tee shot sailed over the green into matted, thin grass where the gallery had been standing. Of those, 18 players have won three or more times.