Woods had a 5-foot birdie putt at the 18th hole that would have extended the match, but he narrowly missed it the right.
On a day when Woods played solid from tee to green, the missed putt on the final hole told the whole story of his round.
"I didn't miss a single shot coming in, which is good," Woods said. "And that was fun to hit the ball that well. Unfortunately I just didn't make a putt when I needed it."
After his 1 up victory over Gonzalo Fernandez-Castano on Wednesday, Woods had complained that he was having trouble reading the greens. On Thursday, his troubles on the grainy Bentgrass surfaces started on the first hole, where he missed a 13-footer for birdie. From there his struggles on the greens would plague the rest of his round.
"I was fighting the blocks all day with my putter," Woods said. "The putter was going back a little bit shut, and subsequently I let it go and it goes left, and I block it right.
"I'm taking it shut going back. I need to make that toe move. And I need to feel the release of my stroke. And it's hard to release it when the blade is going under. It's shut. And hence I block it open."
It was the third straight time that Woods, a three-time champion in this event, failed to get out of the second round.
Two weeks ago, Woods couldn't buy a putt at Pebble Beach and closed with a 75, 11 shots worse than playing partner Phil Mickelson. He also struggled to make putts in Abu Dhabi, when he failed to win despite being tied for the 54-hole lead with Robert Rock.
Watney will face Lee Westwood in the third round.
"I'm very happy to move on. I feel a bit fortunate, as well," Watney said. "We don't see him miss putts like that very often. And there were a few of them."
Westwood had no trouble against Robert Karlsson of Sweden, advancing to the third round for the first time in 12 tries at this fickle event.
"Need more clothes. Didn't pack for long enough!" Westwood jokingly tweeted.
The surprise was that Scotland had two players remaining -- former British Open champion Paul Lawrie took down Ryo Ishikawa, and Martin Laird defeated Matteo Manassero. Lawrie and Laird face each other in the third round.
In other matches Thursday:
• U.S. Open champion Rory McIlroy, the No. 2 seed, made only three birdies but won two straight holes with par to put away Anders Hansen and advance to the third round. He plays Miguel Angel Jimenez, the 48-year-old Spaniard who beat PGA champion Keegan Bradley.
• Dustin Johnson, headed for defeat in the opening round until outlasting Jim Furyk in 20 holes, blasted Francesco Molinari early and rolled to a 7-and-5 win. "I was definitely in a better mood," Johnson said of the short day.
Johnson has played two medium-length players in Furyk and Molinari. Next up is another pea shooter, Mark Wilson, who breezed to a win over Robert Rock of England.
• Steve Stricker celebrated his 45th birthday in style. Two down on the back nine, he rallied to catch Louis Oosthuizen, then won the match with a 20-foot birdie putt on the 18th hole that had so much break even Stricker wasn't sure he could make it.
Seven Americans, seven Europeans, an Asian and an Australian (John Senden) represent the 16 players left.
Neither Watney nor Woods putted well on their opening nine holes, where they each had only one birdie. After nine holes, they were all square.
At the 10th hole, Woods hit a 3-metal off the tee into the right fairway bunker. With the ball on a downslope, Woods hit his next shot right into the cacti, near the green. The errant shot from the bunker sent him into a fury as he let out a barrage of expletives, culminating in him angrily slinging his club toward his golf bag. He cursed under his breath as he walked down the fairway.
Woods would end up making his second straight double bogey on the hole. That outburst spoke to the frustration that he felt on the Ritz-Carlton greens in both of his matches.
Watney never lost the lead after the 10th. Woods would get back to 1 down heading into the 18th and when his 8-iron approach from 185 yards finished five feet from the hole, extra holes seemed certain.
Watney turned his yardage book to the first hole pin sheet in anticipation of Woods making his putt. But it wasn't to be.
"The old adage is to expect your opponent to make it," Watney said. "And when it's Tiger Woods, you really expect him to make it."
"I hit the ball well all day today," Woods said. "And unfortunately I just did not make enough putts to extend the match or even win the match."
Woods will play the Honda Classic next week in Palm Beach Gardens, Fla. In between now and then he'll have a few days to try to cure his putting woes. But an ever-optimistic Woods has a shorter time frame to repair his ailing stroke.
Said Woods: "I should be able to fix it in about a day."
Information from The Associated Press was used in this report.