The world number eight followed up a record-breaking opening round with a disastrous 78 on Friday, needing 40 shots to escape the back nine.
The Holywood man, whose run of poor Friday performances after excellent opening rounds continues to be a mystery, had recovered from a three-putt bogey on the first with four birdies to lie just three off the lead on three under.
In what was a steady third round performance McIlroy picked up a shot at the third before back-to-back birdies at five and six saw him move to 2-under.
A fourth birdie followed on the sixteenth, and McIlroy parred his final two holes to complete a 68 and finish three under.
"I didn't get off to a great start, being one over through two holes when you are looking to be one under, but after that I played very solidly and did not put myself in too much trouble," McIlroy said.
"I've shot 64 and 68 on this course which are two really good scores. Yesterday was just one of those days when nothing went right and I couldn't get any momentum.
"It would be good to shoot another good round tomorrow, but I've seen enough in my game to give me confidence going into the Open."
Elsewhere Darren Clarke is on 1-under after a third round 70, while Gareth Maybin's 2-under round of 69 leaves him on level par heading into Sunday's final round.
Despite being the best-performing local player heading into Saturday's third round Michael Hoey endured a difficult afternoon, carding a second successive 3-over round of 75 to leave him on 2-over.