With the East Lothian links course playing fast and furious under a hot sun, the 31-year-old from New Delhi, who missed the cut in his only previous Open in Hoylake in 2006, birdied the first three holes for a dream start.
He carded six-under off the first eight holes to take the lead but slipped to tied second position at four-under after 13 holes at the time of going to press.
Going into the prestigious tournament, Kapur had claimed that he had not set himself any specific target but was keen on playing out the four rounds.
McIlroy, meanwhile, won five times in 2012, including a second major title by eight shots in the US PGA Championship, but has posted just one top-five finish in 2013 since his controversial multi-million pound switch to Nike.
The 24-year-old confessed to feeling “a little lost” as he missed the cut in the Irish Open last month and again cut a downcast figure after he completed the back nine in 42 today, including putting off the 15th green into a bunker.
“I don’t know what you can do,” he said. “You’ve just got to try and play your way out of it. But it’s nothing to do with technique. It’s all mental out there. And then I just need to concentrate, obviously.
“But sometimes I feel like I’m walking around out there and I’m unconscious. I just need to try to think more. I’m trying to focus and trying to concentrate. But I can’t really fathom it at the minute and it’s hard to stand up here and tell you guys what’s really wrong.”
McIlroy walked off the course midway through the Round 2 of his defence of the Honda Classic saying he was not in the “right place mentally” and threw one club in frustration and leant on another so hard that he bent the shaft out of shape in the final round of the US Open.