Fred Couples carded a 3-under-par 68 and shares fourth place with European Ryder Cuppers Padraig Harrington and Miguel Angel Jimenez. Chris DiMarco, who at one point was 6 under par, and Jay Haas are one stroke further back at minus-2.
Woods opened with a birdie at the par-4 first, but tripped to a bogey at the next at Sherwood Country Club. He then converted back-to-back birdies from the par-4 fourth.
The world No. 2 cruised to the back nine with four straight pars before draining a birdie putt at the par-4 10th to get to 3 under.
Woods stumbled to his second bogey at the par-3 12th, but battled right back with a two-putt birdie at the par-5 13th to get back to 3 under. He then dropped in a 15-foot birdie putt at 17 to grab a share of the lead.
'I played alright today. I wouldn't say I played great,' said Woods, who won here in 2001 and is a three-time runner-up. 'I hit a couple of bad shots here and there, but I hit it close enough where I made enough birdies, but also felt like I didn't really putt well today.'
Furyk, the 2003 U.S. Open champion, got off to hot start with birdies at the second and third. He sank a birdie try at the par-4 seventh and again came right back to birdie the next to move to 4 under.
The 34-year-old stumbled to a bogey at the par-3 12th. For the third time, Furyk made consecutive birdies, this time from the par-5 16th to climb to 5 under. He looked to be the overnight leader, but faltered to a bogey at the last after missing the green.
'The golf course is playing real wet out there. Guys could be pretty aggressive,' said Furyk, who is playing in this event for the third time. 'So, it's good to get out there and get off to a good start early and keep it going.'
Montgomerie began his round with a birdie at the second. He came back to birdie No. 4. The Scotsman birdied the par-4 ninth to move to minus-3.
The 28-time winner on the European Tour ran home a 5-foot eagle putt at the 16th to jump into the lead at 5 under. However, his tee shot flew the green at the next and plugged into a steep slope.
After a drop, Montgomerie pitched back through the front edge of the green. He got up and down from there for bogey and a share of the first-round lead.
'I got to 5 under which is good,' Monty said. 'I hit a 5-wood into about 5 feet at the 16th for eagle. And I made a big, silly mistake on the 17th. Four under is okay though. Anytime you equal Tiger Woods, well, it's okay.'
Chad Campbell, Stewart Cink and defending champion Davis Love III share ninth place at 1-under-par 70. John Daly, Todd Hamilton and Kenny Perry are three strokes further back at plus-2.
World No. 1 Vijay Singh played the first four holes at minus-4, but struggled the rest of the way. He carded a 3-over-par 74 and shared last place in the 16-player field with Stephen Ames.