The Northern Irishman missed a good birdie chance at the last and settled for a five-under 66, a score matched by Dahmen, who was bogey-free.
They headed a large group including American Patrick Reed by one stroke, while defending champion Jason Day of Australia was two back.
Two days before his 30th birthday, McIlroy did not quite fire on all cylinders, but managed to do what he says has elevated his performances this year -- score well without being at the top of his game.
"I didn't play my best today, but I managed my game well, scrambled well (and) it added up to a good score at the end of the day," he told reporters.
McIlroy's second shot from the right rough at the par-four 12th, where he was stymied from taking dead aim at the pin, had the gallery buzzing.
He threaded his ball between two trees, hitting a deliberate hook that could hardly have been better executed.
His ball skirted a greenside bunker before taking the slope and rolling deliciously down to four feet. Never mind that he subsequently missed the birdie putt.
"I probably didn't deserve to make the putt after the tee shot," McIlroy said of the three-wood that got him into trouble in the first place.
"I said to my caddie Harry (before the second shot): 'Is left bunker OK?'
"That's where I assumed it was going to go but it came out perfect, had a little more cut on it than I thought it would out of the rough, took a nice couple of hops and got up there pretty close."
McIlroy was the best player in the world for the first three months of 2019 but could hardly have picked a worse place than Augusta National for his shoddiest performance of the year.
His fifth chance to complete the career grand slam ended with a tie for 21st at the Masters, but two weeks off gave him a chance to analyse his performance.
He concluded that he had fallen into some minor technical faults with his driver.
Close behind McIlroy and Dahmen was 2018 Masters champion Reed, who said it had been important to take advantage of the perfect morning conditions.
"The biggest thing, especially around this place is to play boring golf," he said. "I was able to put the ball from point A to point B fairly well and make some putts."
(Reporting by Andrew Both in Cary, North Carolina, editing by Ed Osmond)
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