Woods overcame a shaky start to make three birdies before putting up a seven at the par five 15th en route to an even par 72.
"I didn't really get anything going today," Woods told reporters. "I was just kind of plugging along.
"All of us were watching Rosie get things going on the front nine."
After opening with a 75, Rose, who played with Woods, turned his day around by going 10-under in a 12-hole stretch that included two eagles and six birdies. He finished the day three shots off the lead on six-under 138.
"I guess the two eagles on the front kind of kick started everything," world number three Rose said. "I kept just pushing myself to keep trying to go as low as I could.
"Once you're through the cut line, then you've got to build your way into the tournament. It was a great day."
Former world number one Kaymer is looking for his first PGA Tour victory since 2014 when he won the U.S. Open. The 34-year-old held the outright lead before a bogey at the last.
Lee was bogey-free with five birdies and the early finishing Merritt mixed an eagle with six birdies and two bogeys.
Rory McIlroy (71) missed the projected cut line of one over by a stroke.
"I was cruising and made that bogey on 15," said the Northern Irishman, who had another bogey at the 17th before collecting a birdie at the last to fall just short.
Fed Ex points leader Matt Kuchar (74) missed by two, while Jason Day (74) and Phil Mickelson (79) were five-over par.
Mickelson opened with a triple bogey and added a double and five bogeys to match his worst round at Muirfield Village.
Day had six bogeys and a double.
(Reporting by Gene Cherry in Raleigh, North Carolina; Editing by Peter Rutherford)
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