That put the 23-year-old Asian joint top at the tournament hosted by Jack Nicklaus, level with American Bo Van Pelt, who birdied six of his first nine holes before losing some momentum after the turn.
"I hit the ball very well today, had a lot of birdie putts and putted well so I'm very happy with the round," Matsuyama, speaking through an interpreter, told Golf Channel.
Scotland's Russell Knox and American Jason Dufner carded 66s while world number two Jordan Spieth, the highest-ranked player in the field, recovered from a slow start to card a 68.
Woods, however, covered his first nine holes in an ugly four-over 40 before scrambling on his second nine with three late birdies to salvage his round.
"Some of the shots I hit were really, really good, but then I also had some really bad shots," the former world number one said. "And we need to work on that, and eliminate the bad ones."
Woods, a five-times champion at Muirfield Village whose world ranking has plummeted to 172nd, ended his roller-coaster round with five birdies, four bogeys and a double at the tricky par-four 18th, his ninth hole of the day.
He is playing only his fifth PGA Tour event of the season after struggling with his game, and has not competed since tying for 69th at the Players Championship last month.
Masters champion Spieth, already a double winner on the U.S. circuit this season, was delighted with his score after teeing off at the 10th.
"It's not easy to start on number 10, it's a brutal hole," said the 21-year-old American. "Great up-and-downs the first four holes before I finally hit a wedge in there tight on 14 and a good up-and-down out of the bunker on 15 for birdie.
"To shoot two-under on the back nine is a great score. Most of the chances are on the front. Very pleased with the round."
(Reporting by Mark Lamport-Stokes in Los Angeles; Editing by Larry Fine)
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