In addition to winning here a year ago, Fowler led at the halfway mark of the 2016 Honda Classic before fading over the final two rounds. He left his shot out of the bunker short into the fringe. Woods is T-20 alongside 15 others.
He answered with a birdie, made par the rest of the way and shot 70.
"It was a pretty tough day", McIlroy said.
He's not flashy and generally plays in the shadow of fellow players Dustin Johnson, Jordan Spieth, k and Ricky Fowler, but Justin Thomas is quietly putting together another strong and consistent season.
Rickie Fowler - Again, I usually go for longer shots in these type of articles, but it's hard not to include Rickie Fowler with +800 odds.
Simpson turned in an equally impressive round during the afternoon wave.
"I think we need to honor those who passed away but I think we also need to honor those who made it through", said Sandra Hicks, a volunteer at The Honda Classic.
"I'm very pleased", Woods said.
The Champion Course here at PGA National is the kind of course that magnifies misses and exposes a player if he's slightly off with his game. Playing in the group ahead of Woods, Thomas reached 4-under before a bogey on No. 6. Once he got there, he figured his best chance at birdie was to hit 3-wood on or near the green.
Simpson is coming off a missed cut at the Waste Management Phoenix Open in early February. His best result came in 2012, when he finished second overall, and obviously since then he's struggled in a very general way.
Beginning the tournament on the back nine, Woods reached 2-under with birdies on two of his first four holes.
Woods had only one big blunder - a double bogey on the par-5 third hole when he missed the green and missed a 3-foot putt - in an otherwise stress-free round. It took a decent effort to hole out for a double bogey having tried to be overly cute with a bunker shot and being spooked by a gust of wind as he attempted what should have been a tap-in for a six.
The Irishman repeated that feat on the back nine, finishing with two birdies to cancel out bogeys on the 11th and 16th holes.
"It was just my seventh round (of 2018), so easily it was my best ball-striking day", he said. "I wasn't very good on Friday, more importantly, I didn't putt very good". Even one of his two wild drives, when his ball landed behind two carts that were selling frozen lemonade and soft pretzels, he still had a good angle to the green.
NOTES: Danny Lee withdrew on Thursday due to undisclosed personal reasons.