Michael Campbell, the U.S. Open winner, also played poorly down the stretch and is in third at 1-over-par 73.
Vijay Singh never fully recovered from a quadruple-bogey 7 at the 11th and shot a 3-over-par 75, which is good for fourth place.
This two-round tournament is reserved for the 2005 PGA Tour's major championship winners. Since Woods won both the Masters and British Open, Singh got into the field based on his performance in the four majors.
Everyone struggled early in the first round, as both Woods and Singh bogeyed the first. The wind was howling, but the quartet made birdie at the par-5 second.
Woods continued to play poorly early on, as he three-putted from 12 feet for a bogey at the third. Mickelson became the first player to reach 2 under par as he rolled in an 8-footer for birdie at the fourth.
Woods and Singh collected birdies at the par-4 fifth as both converted putts of less than 5 feet. Campbell made a routine par, but Mickelson made a mess of the hole. The two-time major winner hit a massive drive, but chunked his wedge short of the green. Mickelson chipped to 4 feet and two-putted for the bogey.
The wind was directly into the players at the par-5 sixth and Woods was the only one to make birdie. He came up short with his second, but hit a wedge to 6 feet and sank the birdie putt. That birdie knotted the foursome at 1 under par.
Mickelson ascended up the leaderboard starting at the par-3 seventh. He ran home an 18-foot birdie putt to move to minus-2, but Woods and Campbell went backwards. Woods missed the green, then missed a 12-footer for the save. Campbell three-putted from 65 feet for bogey.
Mickelson and Woods both birdied the eighth as Mickelson hit a sand-wedge to 10 feet and Woods drove the green at the par-4 hole. Woods two-putted from 90 feet to get under par for the event.
All four parred the ninth, but Woods and Campbell made up ground at No. 10. Woods holed a 20-foot, right-to-left breaker for his birdie, while the U.S. Open champion kicked in a 2-footer.
The par-3 11th proved to be a crucial hole, as the international duo was all over the place. Campbell knocked his tee ball into a greenside bunker, but Singh would have gladly taken that spot. He hit his ball into the water on the left, then did it again from the drop zone. Singh missed a 9-footer after he finally reached the green, so he settled for a quadruple-bogey. Campbell made a pretty decent bogey from a buried lie in the trap, while Mickelson made a par.
Woods sank a 12-foot birdie putt at 11 to join Mickelson at 3 under par.
Singh bogeyed 12 and 13 to fall to plus-5, eight shots behind the co-leaders. Woods and Mickelson both missed makeable birdie tries at 13, but remained two ahead of Campbell.
All four players made birdie at the par-5 14th, although at least one player squandered a solid opportunity for eagle. Campbell left himself with 4 feet for a three, but missed the putt and settled for a birdie.
Campbell bogeyed 15 from behind the green, then the players stepped up to the 501-yard, par-4 16th, which was playing into the wind. Mickelson and Campbell both pulled fairway-metals into a hazard with their second shots, while Woods and Singh safely reached the long green. Mickelson pitched his fourth after a drop to 3 feet and Campbell hit his fourth to 10 feet. Woods and Singh two-putted for par and Mickelson and Campbell both made their bogey saves.
The 17th provided more problems for Mickelson and Campbell. Both ended up short of the green, although Campbell's ball bounced off a wall to land in that spot. Mickelson's pitch barely advanced to the green and Campbell chipped to 6 feet. Mickelson missed his 10-footer for par and Campbell missed for his third consecutive bogey.
On 18, Woods had a tough lie and elected to lay up with his second. The decision paid off as he played his third to nine feet and converted the birdie putt.
Woods seemed more happy the round was finally over. Early on Tuesday, Woods had to be carted off the course as he felt extremely ill. He got through it and is in position for title No. 6 in this event.
'It was a very interesting round of golf today,' said Woods, who won last week's Dunlop Phoenix in Japan. 'I survived it. I was trying to not have to run to the bushes. That wasn't very good.'
Singh also made birdie and Campbell had a great look at birdie from 8 feet, but his putt did not fall.
Mickelson hit his second in the water at the closing hole, but pitched his fourth shot to 2 feet. His par putt rolled in the cup, but he is three behind in his bid to successfully defend his title.
If anyone has a good chance, it's probably Mickelson. He fired a 59 in the final round last year to win by five over Singh.
'I just feel rusty,' said Mickelson, who has not played competitively in six weeks. 'Tomorrow, it doesn't feel like a 59 is coming out. It just doesn't feel like it.'