Tiger Woods also lost out in his game against Charles Howell III, making it the first time since 2002 that the world top two players have lost in the first round of the competition.
Despite Lowry bogeying three holes in a row early on, and McIlroy going two shots ahead, the back nine saw a change of fortunes - including two birdies and an eagle for Lowry.
And the eventual match winner admitted that, while he's not getting ahead of himself, he knew he could defeat the world number one.
"It was always going to be a tough match for me against Rory," Lowry told the World Golf Championships website.
"But it's important not to get too high now, because it's only the first round."
This is match play and I got a few breaks where I needed them - I'm just fortunate to win.
He added: "Deep down, I knew I could beat him. I'm not here for no reason. I'm not here to make up the numbers.
"I was quite nervous starting off. I handed him a few holes early doors, but he handed me a few back later on in the round."
Portrush's Graeme McDowell found himself in a closely contested battle with Dubliner Padraig Harrington, with the 15th hole proving key when G-Mac matched his opponent's birdie to stay even.
"That kind of got me buzzing and I made some nice swings coming in," he said.
Harrington's struggles late on, losing two of the last three holes, helped McDowell to the win and sees him go on to face Swede Alexander Noren in the second round.
Among the rest of the action at Dove Mountain, there were wins for the likes of Team Europe's Ryder Cup star Ian Poulter and world number three Luke Donald.
Lee Westwood lost out in his game against Rafael Cabrera Bello over 19 holes, despite the Spaniard lingering 52 spots lower in the world rankings.
"Obviously the odds were in his favour -but in match play, anything can happen. I just tried my best," Cabrera Bello said.
The first round action had been delayed due to heavy snow in the desert area.