Norrman carded a flawless closing 65 at The K Club to finish 14 under par and a shot ahead of 54-hole leader Hurly Long, with former champion Shane Lowry part of a four-way tie for third on 12 under.
McIlroy began the final round two behind Long and birdied the first and fourth either side of a 90-minute weather delay to share the lead, but found water on the seventh, 11th and twice on the 16th as he stumbled to a 74.
Norrman started the day six behind Long but made six birdies in the space of eight holes from the seventh and then two-putted the par-five 18th for another to set the clubhouse target on 14 under.
Long was briefly on the same score thanks to a birdie on the 12th, but bogeyed the next two holes and was unable to make an eagle on the last to force a play-off.
“It means everything,” Norrman said after his second DP World Tour title in two months, the first coming in the co-sanctioned Barbasol Championship in July.
“It’s such a cool event to play, the fans have been incredible and I really had a fun week out there. It was a long wait (for everyone to finish) but I’m super happy. I can’t believe this happened. I’m over the moon.
“I was starting off a little so-so and I made two birdies and got a rain delay, had a coffee and then went out and flushed it.
“I knew there were a few chances out there. I didn’t really have a number in my head. I just tried to press on with some birdie looks and saved a few great pars too to keep the momentum going.”
McIlroy, who lifted the trophy the last time The K Club hosted the event in 2016, followed his early birdies with a double bogey on the seventh and a bogey on the 11th before reviving his chances with a birdie on the 13th.
However, the world number two then hit his second shot on the 16th into the water for the second day running and amazingly flew his fourth shot from the drop zone over the green and into the same hazard, eventually getting up and down from a greenside bunker for a triple-bogey eight.
Lowry, who began the week facing questions about his Ryder Cup wild card after recording just one top-10 finish in 2023, carded a third 68 of the week which featured a tap-in eagle on the fourth and four birdies.
“It’s obviously a tournament that I would love to win again,” Lowry, who was still an amateur when he won in 2009, said.
“I’m a little disappointed now because to be honest I didn’t see a scoreboard for a long time out there. There was not many out there and 13 under was still leading when I got to the 16th green.
“I did my best and it’s been a great week. The crowds were incredible. It’s one of the best Irish Opens I can remember.
“All in all it’s been a good week for me and a step in the right direction and I’ll be back to defend my (BMW PGA Championship) title at Wentworth next week, which is exciting as well.”