Rahm, ranked No. 11, showed impressive form on the links two weeks out from the British Open, rolling in six birdies and an eagle on a low-scoring day at Portstewart. He was tied for third place with Englishmen Matthew Southgate and Oliver Fisher.
Daniel Im of the United States, ranked No. 542, and Benjamin Hebert of France, ranked No. 254, held the lead after shooting bogey-free 64s.
McIlroy, the defending champion and part of a heavyweight group containing Rahm and Hideki Matsuyama, parred his last 11 holes and was even par at a tournament which benefits his own foundation. The No. 2-ranked Matsuyama shot 67.
Rahm has taken a break from the PGA Tour, where he is enjoying a breakthrough season, to play in Europe ahead of the British Open at Royal Birkdale from July 20-23. He tied for 10th at the French Open last week and is now in contention in Ireland, another event in the European Tour's Rolex Series with a prize fund of $7 million.
He holed a long birdie putt on No. 1 and followed up a bogey and two birdies by rolling in a 20-foot eagle putt on the 7th. He birdied both par fives on the back nine before a final birdie on the 15th.
''My attitude was probably the best it's been all year,'' said Rahm, who is known for his fiery temperament. ''I was positive all day. Kept my routine going. Stayed calm and the result showed how good it was.''
Im, who made six birdies in his opening eight holes, is leading a European Tour event for only the second time in his career. His best finish this year is 14th.
Hebert was the best of the afternoon starters, picking up four straight birdies from No. 13 to join Im atop the leaderboard.