The Northern Irishman started day four ten shots off the lead but after carding his lowest ever round up Magnolia Lane, his clubhouse target of seven under was just three back of Scottie Scheffler.
The World Number One pulled further ahead with two birdies but then four-putted the last for a double-bogey as McIlroy finished the week three shots behind the winning total for his best ever Masters result, which moved him to second in the DP World Tour Rankings in Partnership with Rolex.
He finished his round in style as he holed a stunning bunker shot only to see playing partner Collin Morikawa make his birdie from the same sand, sparking wild celebrations among the patrons.
“It’s what you dream about, right? You dream about getting yourself in position,” he said.
“I wasn’t quite close enough to the lead (but) to play as well as I did and then to finish like this, I mean, it’s just absolutely incredible.
“This tournament never ceases to amaze and that’s as happy as I’ve ever been on a golf course right there. That was an incredible. I’ve never heard roars like that on the 18th green. It was really cool."
McIlroy had birdied the first from nine feet and two-putted the third after driving right up to the fringe but really came alive around the turn.
The 32-year-old holed a 30-foot right-to-lefter from the fringe at the seventh, recovered from a poor tee-shot at the eighth to take advantage of the par five and then chipped in at the tenth before putting his second to ten feet at the 13th to set up an eagle.
A run of four pars looked to have ended his chances but he holed that spectacular bunker shot on the last to give Scheffler something to think about.
I'm getting the hang of this place after 14 years of trying. I'll keep coming back and keep trying
"I got off to a great start today and I just kept it rolling, I knew I needed my best ever score to at least have a chance," he said. "I shot 65 in the third round here in 2018 and I knew I needed to go a bit lower than that.
"I can't ask anymore of myself, I played a really, really good round of golf... it's the most fun I've had on a golf course in a very long time.
"I really just wanted to post a number, that is what was in my head and I just kept thinking to myself all morning, 'why not me?'.
"When I birdied one I just thought this could be it today. I just stayed aggressive and kept hitting good shots and hitting good putts and played smart around Amen Corner - made my pars and moved on - and then finished like that.
"I'm getting the hang of this place after 14 years of trying. I'll keep coming back and keep trying."