McIlroy finally completed the career Grand Slam by defeating Justin Rose in a playoff, becoming only the sixth player to win all four major championships at least once.
The victory came on his 11th attempt at Augusta and 11 years after his last major title at the 2014 PGA Championship at Valhalla.
"Woods said, 'welcome to the club, kid', which was just so cool," said McIlroy, speaking on The Tonight Show hosted by Jimmy Fallon.
"There are only six people who have ever achieved the career Grand Slam in golf, and there are only four of us who are living.
"Gary Player, who is 90 years old, Jack Nicklaus, who is 86 years old, and then Woods and myself, so it's really cool to be a part of something like that."
McIlroy also revealed that music legend Sir Elton John was among those who sent congratulatory messages following the win.
"I was overwhelmed with the people that reached out from all different walks of life, whether it was culture or entertainment," McIlroy added.
"But the one that got me was Sir Elton John, that was incredible.
"I haven't actually been able to connect with him yet. It was his assistant who left me a voicemail and said Elton John would like to congratulate you in person."
To manage the mental pressure of The Masters, the 35-year-old said he turned to watching Bridgerton and reading a John Grisham novel as a way to unwind.
He admitted that doubts had crept in over the years about whether he would ever complete the Grand Slam, despite always trying to approach Augusta with a positive mindset.
"Everyone came up to me and they were like, 'oh, you don't know what you put us through on that Sunday'. I was like, 'how do you think I was feeling?'" McIlroy said.
"Seriously, it was so emotional. I was just thinking, 'thank goodness that is over'.
"It had been 14 years since that, 11 years trying to achieve the career Grand Slam. I go back every year, put my positive hat on, go with a good attitude and every time I leave on that Sunday night and it wouldn't happen.
"You start to think is it ever going to be your time. Did I miss it, have I lost my opportunity? I think there's a good message in there about never giving up, keep coming back, be strong, be resilient. That patience paid off."
Rory McIlroy may have fallen short of winning the Open Championship in front of his home crowd, but he was full of gratitude and emotion for the unwavering support he received.
Rory McIlroy's eagle on 12 - which he described as one of the "coolest moments" of his career - gives him a chance heading into the final round of The Open Championship.
Rory McIlroy believes he coped well with the pressure of playing The Open on home soil, battling through a wayward day off the tee to remain in contention at Royal Portrush.
South Africa's Christiaan Bezuidenhout is one of five players tied at the top of the Open Championship leaderboard after the opening round at Royal Portrush on Thursday.
