Open latest: McIlroy plays down stunning 63
The 21-year-old equalled the lowest round ever in a major with seven birdies and an eagle for a three-shot clubhouse lead.
McIlroy entered the tournament as second-favourite behind world number one Tiger Woods - who is four shots off the top of the leaderboard.
"I think this is probably the most special, just because it's at St Andrews and it's the Open Championship - but I think the 61 was probably slightly better, if I'm honest," he said.
"It's nice to put my name on the few guys that have shot 63 in majors.
"It would have been lovely to shoot 62, but I can't really complain."
McIlroy missed a makeable birdie putt on 17, which would have given him a 62 - a feat never before achieved in a major.
And he admitted he had begun thinking about his final score at that point.
"It sort of went through my mind on 17 that 62 would have been the lowest round in a major. That's probably why I missed the putt," McIlroy admitted.
"Definitely the one on 17 was one I let get away, but it was still a fantastic score.
"I didn't get off to a flying start but I was one under through eight holes and then the eagle on nine (where he holed from 15 feet after driving the green) really sort of turned things around for me and I just got going from there."
Rory said that Graeme McDowell's recent US Open win was a major inspiration for him.
"Graeme's win definitely gave me a lot of belief," said McIlroy. "If he can go out and win a Major the way he did, there's no reason I can't go out and have good chances to win some of the others. I said afterwards that I wouldn't like to be the only Irishman at the Ryder Cup without a Major.
"I always thought a Major was a little farther away than it probably is. I thought I would need another two or three or four years of development in my game to challenge for Majors. But Graeme's win at Pebble Beach made me realise that it might not be as far away as I thought."
In his first round as a Major champion, McDowell birdied the last for a 71, one shot adrift of defending Open champion Stewart Cink.