McIlroy helped set the pace with Tom Kim of South Korea, Bernd Wiesberger of Austria, Rasmus Hojgaard of Denmark and American Patrick Cantlay at five-under par, but Lee was just a shot adrift with six others, including Brooks Koepka, on Thursday.
Lee had six bordies in his 66, and would have been in a tie for the lead too if he hadn't bogeyed the last hole.
World No. 1 Scottie Scheffler made three consecutive birdies to get to three under after 10 holes, but he gave one back at 11 and finished at two-under 68, tied for 27th place.
"You know, I did some good stuff," Scheffler said. "Wish I could have gotten a few more shots out of it but, overall, I think pretty solid."
An uneven start for McIlroy was erased by consecutive birdies and a sand save for par at the ninth, helping shave off two bogeys on his opening nine -- including his first hole of the day at No.10 -- and the 18th. He credited his putting for a brilliant eagle at No. 1, a par-five playing over 600 yards this week.
"I played the par fives well. I think I hit all three greens in regulation: one putt dropped for eagle and two two-putt birdies always helps the card," McIlroy said.
"Overall, good to get my first round of competitive golf on a links golf course and it's obviously a great start to the tournament."
Scotland's Robert MacIntyre, who won the event in 2023, played with McIlroy and defending champion Chris Gotterup to large and noisy galleries. MacIntyre sits two shots back of the lead, tied with US Open champ Wyndham Clark and a dozen others after first-round 67s.
Adam Scott, warming up for his 101st consecutive major championship appearance next week at the British Open at Royal Birkdale, opened up with a one-under 69.
