Dyson was trailing by a shot when he birdied the 17th to draw even. His round of 4-under 67 left him 15-under 269 for the tournament. Dyson finished ninth at the British Open two weeks ago and called his play in the Irish Open "probably the best golf I've ever played."
Green all but handed Dyson the victory when his approach on the 18th led to a three-putt bogey, only his second of the day. He finished at 68 for a 14 under total.
"You always feel sorry for somebody when that happens," Dyson said. "I'd much prefer to win it with a birdie. But I'll take what I can get."
Scotland's Stephen Gallacher was third after a 68 left him at 12 under.
With the victory, Dyson qualified for next week's World Golf Championship in Akron, Ohio. He received the Waterford Crystal winner's trophy from Irish Prime Minister Enda Kenny and $360,000.
More than 20,000 fans lined the fairways Sunday in hopes of seeing fireworks by the Emerald Isle's homegrown stars.
But British Open winner Darren Clarke and three-time-major winner Padraig Harrington didn't even survive Friday's cut. Sunday's two surviving local favorites — Northern Ireland's Rory McIlroy and Graeme McDowell — both had mediocre performances again.
McIlroy closed with a 71 to finish 12 shots behind Dyson.
"It wasn't quite the result I was looking for," said the U.S. Open champion.
McDowell, who won the U.S. Open in 2010, said his round of 70 for a 4-under finish felt "very much like Groundhog Day. These greens just drove me insane all week."
Dyson started the day in a three-way tie for the lead with Green and David Howell. While Howell fell off the pace with narrow misses on long putts, Green jumped to a two-shot lead with four birdies on the front nine.
Dyson caught Green with birdies on the 10th and 11th holes. Then Dyson missed a 6-foot birdie opportunity on No. 12 followed by an even worse 4-foot miss for a bogey on No. 13.
"We were just trading blows," Dyson said, "and then I three-putted the 13th, which was a shame."
Dyson pulled even again by birdieing the par-5 16th hole. Green, in the group behind, shot a stunning approach shot on the 16th, leaving him a 10-foot eagle chance. But his first putt rolled narrowly right; his birdie gave him a one-shot lead again.
Dyson didn't let up, dropping his second shot on the par-4 17th within a few feet of the cup for a birdie and a tie with Green. He parred the 18th and watched from the clubhouse, head in hands, as Green took the 18th tee.
Green's drive came close to landing in a creek. But he found himself with a good line to the cup, and chose his 8-iron for the 150-yard shot into a light crosswind. It dropped far short, barely reaching the green's near corner. Green dropped his head in disappointment. Dyson, watching on TV, distractedly bit his nails.
Green's 60-foot birdie putt for the win went straight to the hole — but cut inches to the right at the last moment and rolled 10 feet beyond. His attempt to send the tournament into a two-way playoff rolled a few inches left.
"I didn't expect the 8-iron on the last hole to come up so short," said the 40-year-old Green, a rare left-hander on the European Tour. "I had a great angle."
Of his long-shot putt for victory, he said: "It was a pressure putt, and I hit it a bit too hard."