McIlroy opened with a 6-under-par 66 that included six birdies. He finished in a tie for 23rd at the U.S. Open last month following an eighth-place tie at the Masters. He's seeking his third career major championship but hasn't done any better than a third-place tie at the British Open.
Italy's Matteo Manassero is alone in second at 5-under. World No. 1 Adam Scott, Sergio Garcia and Jim Furyk, will start the second round at 4-under.
Meanwhile, Woods opened with back-to-back bogeys before getting hot. He birdied five of his last eight holes for a 3-under-par 69 that leaves him three shots back.
Defending champion and five-time major winner Phil Mickelson had a wild ride and finished at 2-over, while Masters champ Bubba Watson had a good start to his first round but finished with a 76.
