Justin Thomas, who won last year's $10 million bonus as the FedExCup champion, carded a two-under 69.
Former world number one Woods, in his first start since a runner-up finish behind Koepka at the PGA Championship this month, mixed two birdies with two bogeys at Ridgewood Country Club as he continued a trend of so-so starts.
Woods, who has broken 70 in the opening round only twice in 15 events in 2018, said less-than-ideal yardages were to blame for his lack of precision with his approach shots.
"I kept having the half-club," said the 14-times major champion, who might need another round of par or better on Friday to make the cut. "I just wasn't quite hitting it close. I just didn't quite have the feel to hit those really soft shots."
Koepka, who also won this year's U.S. Open, was rewarded for his typically aggressive approach off the tee. Starting on the back nine, the highlight of his round was an eagle at the par-five 17th.
"I like being able to hit driver on most holes and if you put it in position you can have some wedges in here and take advantage of your length," he said.
"And I did that pretty well today. A few misses in the rough but these fairways aren't exactly wide so I don't expect to hit them all."
At the 17th hole, Koepka picked up five strokes on playing companion Johnson, who ran up a triple bogey after a wild hook off the tee.
"The triple was just so bad," said Johnson. "I hit that 70 yards left of where I was looking.
"I haven't hit that shot in a long time, so it was kind of funny. Wasn't funny when I made a triple though."
Johnson fought back and almost had a hole in one at the par-three sixth where his ball lipped out of the hole.
Joint leader Lovemark could hardly have wished for a better start with birdies at the first three holes.
"Good way to start, kind of lost it in the middle of the round and then found some good stuff coming in," said the 30-year-old, who entered the playoffs ranked 86th.
The top 125 players in the FedExCup standings were eligible for the first of four playoff events that culminate with a 30-player field for the Sept. 20-23 Tour Championship at East Lake Golf Club in Atlanta.
(Reporting by Frank Pingue in Toronto; Additional reporting by Andrew Both in Cary, North Carolina; Editing by Ed Osmond/Peter Rutherford)
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